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Fast bubble dynamics and sizing
Journal of Sound and Vibration ] (]]]]) ]]]–]]]
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Fast bubble dynamics and sizing Krzysztof Czarnecki a, Damien Fouan b,c, Younes Achaoui b, Serge Mensah b,n a Gdańsk University of Technology, ul. Faculty of Electronics, Telecommunications and Informatics Narutowicza 11/12, 80-233 Gdańsk, Poland b CNRS - Laboratoire de Mécanique et d'Acoustique, 4 impasse Nikola TESLA, CS 40006, 13453 MARSEILLE CEDEX 13, Aix Marseille University, France c BF Systemes SAS, Technopôle de la Mer 229, chemin de la Farlène 83500 - La Seyne-sur-Mer, France
a r t i c l e i n f o
abstract
Article history: Received 15 October 2014 Received in revised form 1 June 2015 Accepted 20 June 2015 Handling Editor: L. Huang
Single bubble sizing is usually performed by measuring the resonant bubble response using the Dual Frequency Ultrasound Method. However, in practice, the use of millisecond-duration chirp-like waves yields nonlinear distortions of the bubble oscillations. In comparison with the resonant curve obtained under harmonic excitation, it was observed that the bubble dynamic response shifted by up to 20 percent of the resonant frequency with bubble radii of less than 100 μm. In the case of low pressure waves (P o 5 kPa), an approximate formula for the apparent frequency shift is derived. Simulated and experimental bubble responses are analyzed in the time–frequency domain using an enhanced concentrated (reassigned) spectrogram. The difference in the resonant frequency resulted from the persistence of the resonant mode in the bubble response. Numerical simulations in which these findings are extended to pairs of coupled bubbles and to bubble clouds are also presented. & 2015 Elsevier Ltd. All rights reserved.
1. Introduction Microbubble detection and sizing methods still remain to be found for a wide range of industrial and scientific purposes. Most conventional approaches focus on the presence of bubbles in liquids [1], which even at low concentrations may considerably impact the thermodynamic behavior of the fluid mixture involved. Oceanic measurements have helped us to understand role of bubbles (mass and energy transfers are bubble size dependent) in greenhouse gas cycles [2–4] and how they affect sound propagation [5]. Recent nuclear power studies have focused on the assessment of undesirable microbubble formation in liquid sodium or liquid CO2 coolants [6–10] which can affect ultrasonic control systems as well as increasing the risk of gas pocket formation (a core safety issue). Bubbles are now being used to reduce scattering cross section of modern submarines and new bubble-filled anechoic coating materials (metamaterial properties) are currently being developed [11]. In other applications, microbubbles are used as bioactive food ingredients, green adjuncts for water or surface cleaning [12–15], ultrasound contrast agents [16–20] and as efficient pumps or activators in the field of microtechnology [21]. In bubbles driven by sound fields with sufficiently high pressure levels, the nonlinearities could be used to assess their size and concentration. Procedures of this kind are being used to grade the bubble occurring in the bloodstream and prevent the risks associated with surgery (cardiopulmonary bypass) induced gas emboli and decompression sickness [22–24]. Introduced or induced
n
Corresponding author. E-mail address: [email protected] (S. Mensah).
http://dx.doi.org/10.1016/j.jsv.2015.06.038 0022-460X/& 2015 Elsevier Ltd. All rights reserved.
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microbubbles can improve (pre-seeding) non-invasive surgery based on high intensity focused ultrasounds [25] or promote the delivery of drugs (sonoporation) [26] with enhanced therapeutic indexes. Nano-droplets of liquid perfluorocabone can be vaporized with ultrasounds, for example, once they have reached extravascular cancer cells thanks to the Enhanced Permeability and Retention (EPR) process. In order to extravasate neo-vasculature [27], the droplets must be less than 400 nm in diameter and then be transformed into micron-sized (ideally between 0.5 and 6 μm) echogenic bubbles. With the Dual-Frequency Ultrasound Method (DFUM) [28,29], resonant bubbles are no longer liable to be confused with larger ones, as usually occurring in the case of linear and harmonic techniques [30,31]. In addition, the DFUM has proved to be efficient even when taken out to the surf zone to measure bubble sizes under breaking waves [2] or when working with a low mechanical index as well in in vitro and in vivo studies [32–34]. The DFUM is based on the fact that the resonance frequency of a bubble is inversely proportional to its radius. There are several ways of implementing DFUM [1]. The combination-frequency approach consists in transmitting two high frequencies: the difference between them is equal to the bubble resonance. This approach requires high level pressure waves of the order of 100 kPa to generate the bubble resonance. Alternatively, the modulation frequency approach requires less pressure (less than 5 kPa) and was therefore used in this study, where linear behavior is required. The frequency f imag of the first wave, the “imaging wave”, is relatively high (High Frequency: HF) in comparison with the resonance frequency f res of the bubble [28], and with the frequency f pump of the so-called “pumping wave” (Low Frequency: LF). As the bubble oscillates under the LF pressure field, the instantaneous amplitude and frequency of the backscattered imaging wave are modulated by the cross-section of the bubble. The oscillations, which are most observable at sum-and-difference frequencies f beat ¼ ∣f imag 7f pump ∣, therefore reach a maximum amplitude when f pump ¼ f res . This procedure provides an efficient means of jointly detecting and sizing microbubbles. Multiple harmonic pumping waves have been recently used to further increase the range of radii of the bubbles characterized [35]. In addition, not only amplitude modulation but also phase modulation of the signals scattered by the pulsating bubbles can be exploited in order to determine their size [36]. In the classical DFUM implementation based on the assumption that the bubble behavior is stationary, sequential detection of mono-disperse bubble sets is carried out by applying stepped frequency bursts. However, these “long sweep duration” procedures are not appropriate for dealing with fleeting or moving bubbles (such as those crossing the tricuspid valve in the right ventricle in less than 20 ms). The aim of the present paper is to present a new signal processing approach based on short-duration excitations (of less than 10 ms) in order to overcome the above limitations. The resonant behavior of a bubble activated by a chirp pumping wave shows a specific pattern of amplitude modulation, which determines the resonance frequency. However, as we will see in the next section, it has emerged that this resonant frequency is affected by the chirp duration, or rather, in a given frequency range, by the chirp rate. In the quasi-linear regime (i.e., at pump pressure amplitudes o5 kPa), a numerical approach is proposed here for quantifying the frequency shift, based on the Harmonic (i.e., no frequency modulation) Resonance Frequency (HRF). The analytically estimated bias matched the numerical simulations based on the modified Rayleigh–Plesset equation. A dedicated time–frequency transformation was developed using the reassignment technique for use with both numerical and experimental data [37–39]. The high resolution obtained makes it possible to clearly identify the reason for the frequency shift. Predictions of bubble-pair and bubble-cloud responses are also presented.
2. Harmonic versus dynamic bubble resonances 2.1. Harmonic resonance frequency of a bubble The volumetric pulsations of a single bubble, which is assumed to be spherical and small (in comparison with the acoustical wavelength), surrounded by an infinite incompressible medium can be described by the well-known nonlinear Rayleigh–Plesset model. In order to finely simulate the bubble resonant behavior, the three damping mechanisms, corresponding to viscous losses at the bubble wall, sound radiation into the fluid and thermal losses, must all be included in the dynamic model. Assuming that the oscillations show polytropic behavior, the modified form of the Rayleigh–Plesset equation is
3 2
ρL RR€ þ R_
2
¼ P B ðt Þ P 0 P ðt Þ þ
! R_ dP B R 1 c dt c
(1)
where ρL is the fluid density, R the instantaneous bubble radius, P(t) the acoustic driving pressure, P 0 the static pressure in the fluid, and c the sound speed in the liquid. The term P B ðtÞ is the pressure exerted on the liquid due to the pulsation of the bubble: 3ς R_ 2σ R0 2σ 4ηL P B ðt Þ ¼ P 0 þ R R0 R R where R0 is the equilibrium bubble radius,
(2)
σ the surface tension, ηL the fluid viscosity, ς the polytropic exponent.
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The frequency ωHRes that maximizes the amplitude of the far-field pressure at a constant frequency and forcing pressure amplitude is given in [40, Eq. (94)] by
ω ω
2 0ð Þ 2 HRes ð
ω
ωÞ
2β 0 ðωÞ ϵ20 ðωÞ : 2 ω20 ðωÞ 2
¼ 1
(3)
where the damping factor β 0 includes viscous and thermal damping in the case of viscous fluids and ω0 is a parameter which depends on the resonance frequency. 2.2. Dynamic resonance frequency of a bubble In the (quasi-)linear regime, the single degree of freedom oscillator formed by the bubble can be described by a standard differential equation written in terms of the damping factor β and the stiffness parameter κ defined respectively by Eqs. (87) and (88) in [40]. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The natural frequency of the unforced damped oscillator is therefore ωnat ¼ κ ðωnat Þ β ðωnat Þ2 : When the driving force is due to a linear-frequency-modulated wave, the Dynamic Resonance Frequency (DRF), ωDRes , at which the maximum amplitude occurs has been determined. This frequency corresponds to the first zero of the function Θr ðtÞ of the chirp-rate r defined by Eq. (45) in [41]. It is adapted here to the case of bubbles forced to oscillate at low pressure levels ðPðtÞ o 5 kPaÞ: pffiffiffi i Θr ðt Þ ¼ 2Re r ð1 þiÞϑn ðω Þ pffiffiffiffi ω ϑðω Þ (4) π h i pffiffiffiffi R u n2 2 where ϑðuÞ ¼ e u 1 þ ð2i= π Þ 0 eu du is the complex error function and ð1 þ iÞ 2 r
ω ðt Þ ¼ pffiffiffi rt þ ωinit β þ iωnat ;
(5)
ωinit is the initial frequency of the chirp and ‘*’ stands for the conjugation. As we will see in Section 4, the dynamic resonant frequency differs increasingly from the harmonic resonant frequency with the chirp rate. 2.3. Dynamics of a coupled-bubble pair For a given pair, the influence of one bubble over the other corresponds to the interaction between the incident wave, the oscillation-induced radiation and the multiple-scattering due to the mutual reflections. Based on the assumption that the interbubble spacing d does not vary with time, the coupling results therefore in an increase of the inertial (added) mass and a reduction of the damping β1 , and gives rise to a lower resonance frequency ω1 of the bubble-pair system. As far as the symmetric lowest mode is concerned, the expressions write [42] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ η þ δδr ðδ þ δr Þ2 =4 ω0 ðδ þ δr Þ ; β1 ¼ : (6) ω1 ¼ ω0 2ð1 þ η þ δδr Þ 1 þ η þ δδ r where η ¼ R0 =d. δ and δr are respectively the total and radiation damping constants. These parameters can be used to predict the dependence of the resonance frequency on the chirp rate. 2.4. Dynamics of a bubble cloud The oscillations and sound emission of a bubble cloud consisting of a large number of bubbles have been studied extensively [43–48] and it has been established that the collective oscillations of large bubble clouds give acoustic emissions as low as several hundreds of Hertz. At these low frequencies which are well below the lowest bubble resonance frequency, the effects of the bubble distribution are quite small and the speed of sound cc in the mixture therefore depends only on the void fraction α. Within the limits of moderate void fractions (several percent) and isothermal behavior, the cloud resonant frequency ωc and the damping δ have been derived [43]: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cc π u u Re Γ ωc ¼ (7) u A 2t R2 π 2 1 þ 20 A α 4
δc ¼ " A 1þ
cc R20 π 2 A2 α 4
" #
π 2 ρl ωR20
c β þ c Re Γ 4 3P gas 0 c
# (8)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where cc ¼ P 0 =ρl αð1 αÞ is the speed of sound in the mixture, the polytropic exponent is described by the complex function Γ of the driving frequency [40, Eq. (65)]. Please cite this article as: K. Czarnecki, et al., Fast bubble dynamics and sizing, Journal of Sound and Vibration (2015), http: //dx.doi.org/10.1016/j.jsv.2015.06.038i
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3. Materials and methods 3.1. Numerical simulation of bubble responses Numerical experiments were performed on an isolated bubble in a viscous fluid, using the Matlab “ODE45” function to solve the modified Rayleigh–Plesset model (1). Due to the nonlinear mixing of the pump and imaging waves, the linearly frequency-modulated US waves were observed in the side-bands apart from the imaging spectral line and the signal to noise ratios of these virtual measurements were adjusted by adding white Gaussian noise. The resonance frequency was approximated using the iterative procedure defined by Eq. (3). In order to prevent the occurrence of interactions with the pump waves as far as possible, we worked in the upper band and carried out a coherent demodulation before applying any other procedure. The pumping pressure amplitude was held constant at each measurement (P pump was set between 5 and 25 kPa), but the frequency was varied continuously and linearly from 20 kHz to 80 kHz. The imaging pressure wave applied was constant in terms of both its amplitude (P imag ¼ 10 kPa) and its frequency (f imag ¼ 1 MHz). The time-dependent changes in the radius were sampled at a frequency of 18 MHz. 3.2. Time–frequency analysis of bubble responses The behavior of the forced bubble can be characterized by taking the energy distribution of its oscillations as a function of the time and the frequency. The use of chirp-controlled waves in the Dual Frequency Method naturally leads one to perform a chirplet decomposition (quasi-matched filtering) in order to obtain a relatively high energy concentration. The envelope of the analyzing window used is described by the Blackman–Harris function: hc ðtÞ ¼ hðtÞexpðiπ rt 2 Þ;
t A ð T h =2; T h =2Þ;
(9)
where r is the chirp-rate and T h is the absolute window width. The input signal s(t) is decomposed into the chirplet transform Sðt; ωÞ as follows: Sðt 0 ; ω0 Þ ¼
TZ h =2
sðτÞhc ðτ t 0 Þexpð iω0 τÞ dτ: n
(10)
T h =2
In order to increase the signal to noise ratio in the resulting transform, the chirp rate value r chosen is equal to the slope of the instantaneous frequency of the pumping wave. To prevent the occurrence of induced cavitation in supersaturated media, low Mechanical Index (MI) conditions (MI o 0:3) are maintained. Under these conditions, the sideband relative (to imaging wave power) power density measured is between 20 and 30 dB. The time–frequency (TF) bubble response analysis involves two-steps. In the first step, the discretized TF complex spectrum S½l; k is multiplied (term by term) by a 2D mask matching the pumping wave in order to reduce most of the noise energy. The inverse chirplet transform is then applied in order to synthesize the corresponding “noise-free” signal. In the second stage, a Short-Time Fourier Transform (STFT) calculated using a narrow Blackman–Harris window (and its derivatives) gives local estimates of both the Channelized Instantaneous Frequency (CIF) and the Local Group Delay (LGD), which are directly used in the reassignment procedure to obtain a concentrated spectrogram [37–39]. A block diagram of the algorithm is presented in Fig. 1. 3.3. Experimental set-up The experiments were performed in a 2 m 3 m 0.6 m water tank in order to mitigate the potential effects of any standing waves. Microbubbles with radii ranging from 10 μm to 200 μm were generated using a hydrojet (Oral B Oxyjet 1000, Braun). A single bubble was then isolated on a 80-μm diameter wire. The advantage of using a single tethered bubble is that it prevents the occurrence of Doppler effects. The dynamics of the tethered bubble differ from those of the free bubble mainly by the dependence of the inertial (or added) mass on the contact angle [49–51]: the resonant frequency may be lowered down to 20percent. However, it was checked that in the case of a bubble with a radius ranging between 40 and 160 μm, no adhesion effects are visible in the volumetric vibrations as long as the bubble is not masked by the wire [35]. The specific measurements presented in appendix on the same tethered and then released bubbles confirm that the resonance frequency of free bubbles can be quite accurately assessed when working with bubbles attached to a wire. The size of the bubble was monitored optically using an immersed optical device comprising a 5 megapixel camera (TXG 50 Baumer) equipped with a telecentric lens having a magnification of 1.5 70 and a depth of field of 660 μm (resolution 2.3 μm per pixel). Two confocused 1-MHz frequency transducers (Imasonic, 1 MHz, f ¼ 90 mm) were used both to emit the imaging waves and to detect the corresponding echoes. To pump the bubble, a low frequency transducer was added (Ultran, GMP 50 kHz). The two transmitters were synchronized using an arbitrary waveform generator (LeCroy, ArbStudio 1104, four channels). The receiver was connected to an oscilloscope (Agilent Technologies, InfiniiVision DSO5014A, 100 MHz). The complete set-up is described in Fig. 2. A suitable pumping sweep was chosen for sizing bubbles with radii ranging between 40 μm and 160 μm corresponding to Minnaert frequencies ranging from 20 kHz to 80 kHz. The acoustic pressures at the focal point had been previously measured using low and high frequency hydrophones (Reson, Ref. TC 4034, Please cite this article as: K. Czarnecki, et al., Fast bubble dynamics and sizing, Journal of Sound and Vibration (2015), http: //dx.doi.org/10.1016/j.jsv.2015.06.038i
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Fig. 1. Two-step bubble analysis: after a noise filtering step, the dynamic resonance frequency assessment was further enhanced by applying the reassignment method.
Fig. 2. Schematic diagram of the release and sizing chain.
Precision Acoustic Ref. 1524 & DCPS 060). The maximum acoustic pressure at the focal point in the high frequency field amounted to 10 kPa and in the low frequency field to 3 kPa. 4. Results and discussion 4.1. Temporal bubble response analysis The time responses of a 64-μm bubble are illustrated in Fig. 3 in the case of linear chirp waves of various durations (100, 10 and 1 ms) at a constant pressure P p ¼ 5 kPa. When the signal duration is reduced, less energy is obviously transmitted to the bubble. Please cite this article as: K. Czarnecki, et al., Fast bubble dynamics and sizing, Journal of Sound and Vibration (2015), http: //dx.doi.org/10.1016/j.jsv.2015.06.038i
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Fig. 3. Time responses of a 64-μm bubble to short-duration up-chirp excitations (100, 10 and 1 ms). Pumping pressure level¼ 5 kPa.
Fig. 4. Time responses of a 64-μm bubble to short-duration down-chirp excitations (100, 10 and 1 ms). Pumping pressure level¼ 5 kPa.
Therefore, as shown in Fig. 3, during the swept-sine excitation, the maximum response of the resonance is lower than the maximum response of the harmonic excitation approximated here using the long 100-ms signal. In addition, as the chirp-rate increases, the ratio between the time required to reach the maximum and the signal duration also increases: this value is therefore not reached when the instantaneous frequency of the linear chirp equals the static (harmonic excitation) resonance frequency: 49 kHz in the case of a bubble with a radius of 64 μm (Fig. 3). Fig. 5 shows that the frequency shift depends not only on the chirp-rate but also on the bubble size. In practical situations, the chirp-rate is predetermined, and the Dynamic Resonant Frequency versus the bubble radius can therefore be predicted using (4). This equation can also be applied to down-chirp excitations as shown in Fig. 6 where the bubble ringdown is quantified and shows a quasi-symmetrical shape as compared with Fig. 5. Indeed, it has been established that in the absence of damping, during both run-up and run-down processes with equal absolute chirp-rates, the frequency shifts are the same [41] (Figs. 3–6). In addition, as depicted in Fig. 8, the resonance frequency decreases when the amplitude of the pumping wave increases due to the (nonlinear) softening behavior of the bubble system [52]. The frequency shift is actually more pronounced when linear down-chirps are used as shown in Fig. 9. An efficient method of solving the nonlinear inverse problem (the sizing procedure) might therefore consist in minimizing the least square error between the dynamic resonance frequency measured and that obtained using the Rayleigh–Plesset model (1). In these simulations, the amplitude of the low frequency driving chirp was kept constant across the frequency range. In more practical situations, the amplitude fluctuations in the transducer bandwidth have to be taken into account. The results presented in a previous paper [35] showed that the strong resonance ð5 o Q factor o 30Þ compensates for the amplitude fluctuations in the spanned frequency range and makes bubble sizing possible even outside the transducer 6-dB bandwidth. The frequency shift curves of a 64-μm bubble pair present the same evolution with the chirp rate as a single bubble. The reduction of inter-bubble spacing increases the coupling, which results in a lower damping (Fig. 7). Therefore, the contribution of the natural mode in the system response is relatively more important and thus the ringing-up/down effect is more pronounced than that of a single bubble. Considering the experimental conditions of [45], the frequency shift curves versus the chirp rate of a 1.5-mm bubble cloud show a similar dependence to that of a single bubble. Also we can notice a similar dependence on the respective radii. However, the damping of the cloud depends on the void fraction, which can fluctuate up to several orders of magnitude. In the case of a 5-cm (radius) cloud, with a void fraction ranging between 0.5 percent and 2 percent, the damping (the resonance frequency, respectively) ranges from 180 to 60 (from 600 to 300 Hz, respectively). This could explain the delay in the increase of the frequency Please cite this article as: K. Czarnecki, et al., Fast bubble dynamics and sizing, Journal of Sound and Vibration (2015), http: //dx.doi.org/10.1016/j.jsv.2015.06.038i
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Fig. 5. Normalized frequency shift versus the positive chirp rate with various bubble sizes (linear oscillation regime).
Fig. 6. Normalized frequency shift versus the negative chirp rate with various bubble sizes (linear oscillation regime).
Fig. 7. Normalized frequency shift of 64-μm bubble pairs versus the chirp rate for various interspacing distances.
shift versus the chirp rate in Fig. 10: a high damping factor reduces the contribution of the natural modes in the forced bubble response (see next paragraph), which results in a small perturbation of the off-resonant behavior, and therefore ωDRes =ωHRes C 1. If the chirp rate is to be high however, the resonant phenomenon prevails, inducing a significant frequency shift.
4.2. Time–frequency bubble response analysis Time–frequency distributions can be advantageously used to size microbubbles when the low-chirp rate (r¼0.6 MHz/s) DUFM is used, especially in the presence of noise. In comparison with standard approaches based on the classical spectrogram, the reassignment procedure improves the resolution and decreases the probability of false bubble detection, as shown by the spectral marginals presented in Figs. 11 and 12. These techniques were applied to the experimental characterization of microbubbles tethered to a 80-μm wire (Fig. 13). The results presented in Fig. 14 in the case of a 10-ms signal (r ¼6 MHz/s) show that a clearly visible pattern of energy Please cite this article as: K. Czarnecki, et al., Fast bubble dynamics and sizing, Journal of Sound and Vibration (2015), http: //dx.doi.org/10.1016/j.jsv.2015.06.038i
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Fig. 8. Time responses of a 64-μm bubble to short-duration up-chirp (100, 10 and 1 ms) at a pressure level of 25 kPa.
Fig. 9. Time responses of a 64-μm bubble to short-duration down-chirp (100, 10 and 1 ms) at a pressure level of 25 kPa.
distribution was obtained in the TF domain under laboratory conditions, giving an estimated radius of 38 μm. The bubbles were sized by calculating the resonance frequency from the (classical and concentrated) spectrograms. However, as shown in Fig. 15, short-duration bubble sizing (with a duration of 2 ms) when applying spectrogram techniques results in strong blurring effects, which affects the precision of the results, whereas the concentrated spectrogram calculated from the noise-free signals re-synthesized re-allocates the blurred energy to a coherent location, as can be seen from Fig. 16. This figure shows the dynamic resonance of a 79-μm microbubble excited with a 2-ms signal with relatively high sharpness. Even in the case of high chirp rates (of the order of 30 MHz/s), the corrected DFUM will therefore give relatively reliable detection and characterization performances. On the other hand, the bubble radius estimated using the maximum amplitude approach would have been 73 μm, corresponding to an underestimate of more than 7.5 percent of the equilibrium radius. In addition, the concentrated spectrogram depicted in Fig. 16 shows that the linearity of the frequency modulation (which exhibits a visible bend) is altered at the resonance frequency. Oscillations with a constant frequency superimposed on the volumetric pulsations forced by the pumping chirp persisted for about one-tenth of a millisecond. In the time– frequency domain, the concentrated spectrogram therefore shows a typical Y-shaped pattern which was also clearly obtained in the simulation illustrated in Fig. 17. This bubble response can be interpreted as the superimposition of the free oscillations at the natural frequency (i.e., the impulse response of the bubble), triggered during the passage of resonance, and forced oscillations with an increasing excitation frequency. In practice, this specific pattern should improve the bubble detection and characterization using pattern recognition methods.
5. Conclusion The aim of the present study was to develop short-duration bubble detectors and sizers based on the Dual Frequency Ultrasound Method (DFUM). A fast single bubble characterization was obtained with this method using relatively large chirp-rate pumping waves. These waves are required when short-duration (corresponding to a duration of a millisecond) and/or high repetition rate inspection is performed on moving media such as blood. This method can be used to detect and characterize circulating bubbles in the right ventricle for the prevention of decompression sickness, for example. Please cite this article as: K. Czarnecki, et al., Fast bubble dynamics and sizing, Journal of Sound and Vibration (2015), http: //dx.doi.org/10.1016/j.jsv.2015.06.038i
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Fig. 10. Normalized frequency shift of spherical clouds (radius 5 cm) of 1.5-mm bubbles versus the chirp-rate with various void fractions
Fig. 11. Classical spectrogram of the simulated DFUM signal backscattered by a 64μm bubble and the marginal distributions. The pumping linear chirp frequencies were 20–80 kHz, the duration was 100 ms, and the imaging wave had a central frequency of 1 MHz.
Fig. 12. Concentrated chirplet spectrogram obtained after reassigning the (pre-)processed data. Same simulated conditions as in Fig. 11.
Fig. 13. Spectrogram obtained from an experimentally measured bubble (estimated radius: 38 μm) (with a 10-ms chirp).
Numerical simulations based on a modified Rayleigh–Plesset model have shown that chirp-wave activation leads to a time shift in the maximum amplitude of the bubble response. By analogy with harmonic activation, the dynamic resonance frequency is that of the component which induces the largest oscillations. In the (quasi-)linear regime, the resonant frequency shift, which was found to depend on both the chirp rate and the bubble radius, was quantified numerically.
Please cite this article as: K. Czarnecki, et al., Fast bubble dynamics and sizing, Journal of Sound and Vibration (2015), http: //dx.doi.org/10.1016/j.jsv.2015.06.038i
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Fig. 14. Concentrated spectrogram obtained from an experimentally measured bubble (estimated radius: 38 μm) (with a 10-ms chirp).
Fig. 15. Spectrogram of a 2-ms experimental bubble response. The radius was estimated at 79 μm.
Fig. 16. Same bubble response as in Fig. 15. Reassignment method giving a bubble-radius estimate of 79 μm.
Fig. 17. Concentrated spectrogram of a simulated 64-μm bubble response to a 2-ms chirp wave at a pressure level of 5 kPa. The bubble resonance was characterized by a “Y” shaped pattern of the energy distribution. The harmonic resonant frequency (49 kHz) is clearly visible on the marginal spectrum.
A dedicated TF estimator was also designed to calculate the concentrated spectrogram (based on the reassignment technique) of the noise-free signal obtained using a matched-filtering module. Using this estimator, tethered bubble were experimentally characterized with durations ranging from 10-ms down to 2-ms: the radii of the bubbles detected were in the 40–160 μm range. Under laboratory conditions, the sizing obtained was qualitatively much more accurate than the usual procedure based on the standard spectrogram. Please cite this article as: K. Czarnecki, et al., Fast bubble dynamics and sizing, Journal of Sound and Vibration (2015), http: //dx.doi.org/10.1016/j.jsv.2015.06.038i
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Fig. 18. Follow up of three consecutive images (every 66 ms) of a jumping bubble activated by a 30-kPa 100-cycles burst of frequency equals to the resonance frequency (45 kHz) of the bubble. The bubble is momently pushed away by the radiation force and then attracted to the wire by the secondary Bjerknes forces.
Fig. 19. Image sequence (frame rate 15 Hz) describing the evolution of a 60-μm bubble during the ultrasonic release (below A) and sizing (B) process.
Fig. 20. Comparison between the resonance amplitude modulation curves before (a) and after (b) the detachment of the bubble. The corrected (compensation for the frequency shift) values obtained were 58.9 and 59:9 μm, respectively, in good agreement with the optical measurements (60 μm). The 1 μm difference shows that tethered bubbles are representative of the free-bubble dynamics.
In addition, the concentrated spectrogram makes it possible to keep a good localization in the TF plane, although large up-chirp rates (of up to 30 MHz/s) are used. In the range of bubble radii covered, the sharpness of the concentrated spectrogram made it possible to observe that when millisecond activations are employed, the free response (i.e., the impulse response) of the bubble is excited in addition to the forced response. The observed frequency shift results from the superimposition of these two responses; neglecting this dynamic effect may lead to under-estimating the bubble size by up to 20 percent. The same conclusion applied to pairs and clouds of coupled bubbles.
Acknowledgment This work was partly supported by the French Research Ministry under the “Smart US” project, Ref. ANR-10-BLAN-0311 and MUSEM-2015-IC-STAR project, by the Provence-Alpes-Cte-D'Azur Council, by the Cancrople PACA and by the French Please cite this article as: K. Czarnecki, et al., Fast bubble dynamics and sizing, Journal of Sound and Vibration (2015), http: //dx.doi.org/10.1016/j.jsv.2015.06.038i
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Ministry of Defense, under the “BORA” project. Authors are grateful to Eric Debieu for his contribution in the experimental set up.
Appendix A. Effects of the tethering on the resonance The question of the possible effects of the adhesion to a rigid boundary on the resonance has been previously addressed in [49,50]. It was established that the resonance frequency shows an angle-contact dependence and can be altered by up to 18 percent. However, in our experiments, with bubble radii ranging between 40 and 160 μm, no adhesion effects were visible as long as the bubble was not masked by the 80-μm diameter copper wire. In order to determine the effects of the tethering, measurements were performed on a bubble attached to the wire, and on the same bubble after it had been released and rose due to buoyancy. For this purpose, an ultrasound-driven release process was designed: this process is presented below. Due to surface tension, a bubble in contact with a surface is subject to capillary forces which keep it attached. When a focused high frequency (above the resonance frequency) ultrasound wave is impinging on the bubble, the radiation force pushes the bubble away from its equilibrium position. In addition, because of the pressure gradient, primary Bjerknes forces will tend to move the bubble away from the focus, eventually towards a pressure node. In addition to these forces, secondary Bjerknes forces are induced by the oscillations of the bubble in close proximity to the rigid (and possibly vibrating) wire surface. If the wave frequency chosen is near the resonance frequency, both of the Bjerknes forces will be maximum. Lastly, the buoyancy also acts on the bubble. When an ultrasonic wave hits it, the bubble is pushed away before it enters a steady-state oscillation regime. Once the oscillations have started, they give rise to additional attractive forces towards the wire. The duration of the setting-up of the attractive force is of the order of Q-factor (quality factor) cycles. In the case of a 80-μm bubble, the Q-factor is about 10. If a 100-kPa ultrasonic wave with a central frequency 40 kHz acts on a 80-μm bubble (Minnaert frequency of 40 kHz), the primary and secondary Bjerknes forces will amount to 10 6 N and 10 2 N respectively, the radiation force to 10 5 N and the buoyancy force to 10 8 N. The capillary force cannot be reliably estimated since the contact angle is not visible. This rough evaluation shows that secondary Bjerknes forces are four orders greater than the other forces and need to be minimized. Therefore, 10-cycle pump bursts are used to kick off the bubble without inducing any significant attractive forces. Fig. 18 illustrates how a 100 cycle burst can be used to detach the (jumping) bubble from the wire, and how it is also responsible for its re-attachment. The image sequence (frame rate 15 Hz) presented in Fig. 19 shows the evolution of a 60-μm (optically measured) bubble during the complete sizing process. Once the bubble has been released, the vertical rising trajectory is suddenly shifted to the right at the instant when the pump wave is applied again for the sizing (the transducer was located here on the left side of the image, delay 200 ms). The resonance frequencies can thus be estimated first with the bubble attached (Fig. 20a) and secondly, on the free bubble (Fig. 20b). For this purpose, ten 10-ms chirps are transmitted (chrip-rate¼3 MHz/s), the back-scattering image waves are processed and the amplitude modulation curves are averaged. The corresponding maximum radii amount to 58 μm and 57 μm, respectively. A higher signal to noise ratio can be observed on the amplitude modulation curves of the free bubble, since the focus of the transducer was located several millimeters above the static position of the bubble in order to obtain optimum free-bubble measurements. The relative frequency shift associated with a 60-μm bubble with a 3-MHz/s chirp rate amounts to 1.033. The corrected resonant radii estimated were therefore 58.9 μm and 59.9 μm. These values are in very good agreement with the optical measurements. Reality is certainly rather more complex and we expect greater stiffness and damping to exist due to the adhesion. These parameters could be estimated using greater optical and temporal resolutions. However, in a first approximation, the tethered-bubble dynamics described here seem to be a representative approximation of free bubbles behavior. References [1] T.G. Leighton, From seas to surgeries, from babbling brooks to baby scans: the acoustics of gas bubbles in liquids, International Journal of Modern Physics B 18 (25) (2004) 3267–3314. [2] A.D. Phelps, D.G. Ramble, T.G. Leighton, The use of a combination frequency technique to measure the surf zone bubble population, Journal of the Acoustical Society of America 101 (4) (1997) 1981–1989. [3] H. 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Fast bubble dynamics and sizing Krzysztof Czarnecki a, Damien Fouan b,c, Younes Achaoui b, Serge Mensah b,n a Gdańsk University of Technology, ul. Faculty of Electronics, Telecommunications and Informatics Narutowicza 11/12, 80-233 Gdańsk, Poland b CNRS - Laboratoire de Mécanique et d'Acoustique, 4 impasse Nikola TESLA, CS 40006, 13453 MARSEILLE CEDEX 13, Aix Marseille University, France c BF Systemes SAS, Technopôle de la Mer 229, chemin de la Farlène 83500 - La Seyne-sur-Mer, France
a r t i c l e i n f o
abstract
Article history: Received 15 October 2014 Received in revised form 1 June 2015 Accepted 20 June 2015 Handling Editor: L. Huang
Single bubble sizing is usually performed by measuring the resonant bubble response using the Dual Frequency Ultrasound Method. However, in practice, the use of millisecond-duration chirp-like waves yields nonlinear distortions of the bubble oscillations. In comparison with the resonant curve obtained under harmonic excitation, it was observed that the bubble dynamic response shifted by up to 20 percent of the resonant frequency with bubble radii of less than 100 μm. In the case of low pressure waves (P o 5 kPa), an approximate formula for the apparent frequency shift is derived. Simulated and experimental bubble responses are analyzed in the time–frequency domain using an enhanced concentrated (reassigned) spectrogram. The difference in the resonant frequency resulted from the persistence of the resonant mode in the bubble response. Numerical simulations in which these findings are extended to pairs of coupled bubbles and to bubble clouds are also presented. & 2015 Elsevier Ltd. All rights reserved.
1. Introduction Microbubble detection and sizing methods still remain to be found for a wide range of industrial and scientific purposes. Most conventional approaches focus on the presence of bubbles in liquids [1], which even at low concentrations may considerably impact the thermodynamic behavior of the fluid mixture involved. Oceanic measurements have helped us to understand role of bubbles (mass and energy transfers are bubble size dependent) in greenhouse gas cycles [2–4] and how they affect sound propagation [5]. Recent nuclear power studies have focused on the assessment of undesirable microbubble formation in liquid sodium or liquid CO2 coolants [6–10] which can affect ultrasonic control systems as well as increasing the risk of gas pocket formation (a core safety issue). Bubbles are now being used to reduce scattering cross section of modern submarines and new bubble-filled anechoic coating materials (metamaterial properties) are currently being developed [11]. In other applications, microbubbles are used as bioactive food ingredients, green adjuncts for water or surface cleaning [12–15], ultrasound contrast agents [16–20] and as efficient pumps or activators in the field of microtechnology [21]. In bubbles driven by sound fields with sufficiently high pressure levels, the nonlinearities could be used to assess their size and concentration. Procedures of this kind are being used to grade the bubble occurring in the bloodstream and prevent the risks associated with surgery (cardiopulmonary bypass) induced gas emboli and decompression sickness [22–24]. Introduced or induced
n
Corresponding author. E-mail address: [email protected] (S. Mensah).
http://dx.doi.org/10.1016/j.jsv.2015.06.038 0022-460X/& 2015 Elsevier Ltd. All rights reserved.
Please cite this article as: K. Czarnecki, et al., Fast bubble dynamics and sizing, Journal of Sound and Vibration (2015), http: //dx.doi.org/10.1016/j.jsv.2015.06.038i
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microbubbles can improve (pre-seeding) non-invasive surgery based on high intensity focused ultrasounds [25] or promote the delivery of drugs (sonoporation) [26] with enhanced therapeutic indexes. Nano-droplets of liquid perfluorocabone can be vaporized with ultrasounds, for example, once they have reached extravascular cancer cells thanks to the Enhanced Permeability and Retention (EPR) process. In order to extravasate neo-vasculature [27], the droplets must be less than 400 nm in diameter and then be transformed into micron-sized (ideally between 0.5 and 6 μm) echogenic bubbles. With the Dual-Frequency Ultrasound Method (DFUM) [28,29], resonant bubbles are no longer liable to be confused with larger ones, as usually occurring in the case of linear and harmonic techniques [30,31]. In addition, the DFUM has proved to be efficient even when taken out to the surf zone to measure bubble sizes under breaking waves [2] or when working with a low mechanical index as well in in vitro and in vivo studies [32–34]. The DFUM is based on the fact that the resonance frequency of a bubble is inversely proportional to its radius. There are several ways of implementing DFUM [1]. The combination-frequency approach consists in transmitting two high frequencies: the difference between them is equal to the bubble resonance. This approach requires high level pressure waves of the order of 100 kPa to generate the bubble resonance. Alternatively, the modulation frequency approach requires less pressure (less than 5 kPa) and was therefore used in this study, where linear behavior is required. The frequency f imag of the first wave, the “imaging wave”, is relatively high (High Frequency: HF) in comparison with the resonance frequency f res of the bubble [28], and with the frequency f pump of the so-called “pumping wave” (Low Frequency: LF). As the bubble oscillates under the LF pressure field, the instantaneous amplitude and frequency of the backscattered imaging wave are modulated by the cross-section of the bubble. The oscillations, which are most observable at sum-and-difference frequencies f beat ¼ ∣f imag 7f pump ∣, therefore reach a maximum amplitude when f pump ¼ f res . This procedure provides an efficient means of jointly detecting and sizing microbubbles. Multiple harmonic pumping waves have been recently used to further increase the range of radii of the bubbles characterized [35]. In addition, not only amplitude modulation but also phase modulation of the signals scattered by the pulsating bubbles can be exploited in order to determine their size [36]. In the classical DFUM implementation based on the assumption that the bubble behavior is stationary, sequential detection of mono-disperse bubble sets is carried out by applying stepped frequency bursts. However, these “long sweep duration” procedures are not appropriate for dealing with fleeting or moving bubbles (such as those crossing the tricuspid valve in the right ventricle in less than 20 ms). The aim of the present paper is to present a new signal processing approach based on short-duration excitations (of less than 10 ms) in order to overcome the above limitations. The resonant behavior of a bubble activated by a chirp pumping wave shows a specific pattern of amplitude modulation, which determines the resonance frequency. However, as we will see in the next section, it has emerged that this resonant frequency is affected by the chirp duration, or rather, in a given frequency range, by the chirp rate. In the quasi-linear regime (i.e., at pump pressure amplitudes o5 kPa), a numerical approach is proposed here for quantifying the frequency shift, based on the Harmonic (i.e., no frequency modulation) Resonance Frequency (HRF). The analytically estimated bias matched the numerical simulations based on the modified Rayleigh–Plesset equation. A dedicated time–frequency transformation was developed using the reassignment technique for use with both numerical and experimental data [37–39]. The high resolution obtained makes it possible to clearly identify the reason for the frequency shift. Predictions of bubble-pair and bubble-cloud responses are also presented.
2. Harmonic versus dynamic bubble resonances 2.1. Harmonic resonance frequency of a bubble The volumetric pulsations of a single bubble, which is assumed to be spherical and small (in comparison with the acoustical wavelength), surrounded by an infinite incompressible medium can be described by the well-known nonlinear Rayleigh–Plesset model. In order to finely simulate the bubble resonant behavior, the three damping mechanisms, corresponding to viscous losses at the bubble wall, sound radiation into the fluid and thermal losses, must all be included in the dynamic model. Assuming that the oscillations show polytropic behavior, the modified form of the Rayleigh–Plesset equation is
3 2
ρL RR€ þ R_
2
¼ P B ðt Þ P 0 P ðt Þ þ
! R_ dP B R 1 c dt c
(1)
where ρL is the fluid density, R the instantaneous bubble radius, P(t) the acoustic driving pressure, P 0 the static pressure in the fluid, and c the sound speed in the liquid. The term P B ðtÞ is the pressure exerted on the liquid due to the pulsation of the bubble: 3ς R_ 2σ R0 2σ 4ηL P B ðt Þ ¼ P 0 þ R R0 R R where R0 is the equilibrium bubble radius,
(2)
σ the surface tension, ηL the fluid viscosity, ς the polytropic exponent.
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The frequency ωHRes that maximizes the amplitude of the far-field pressure at a constant frequency and forcing pressure amplitude is given in [40, Eq. (94)] by
ω ω
2 0ð Þ 2 HRes ð
ω
ωÞ
2β 0 ðωÞ ϵ20 ðωÞ : 2 ω20 ðωÞ 2
¼ 1
(3)
where the damping factor β 0 includes viscous and thermal damping in the case of viscous fluids and ω0 is a parameter which depends on the resonance frequency. 2.2. Dynamic resonance frequency of a bubble In the (quasi-)linear regime, the single degree of freedom oscillator formed by the bubble can be described by a standard differential equation written in terms of the damping factor β and the stiffness parameter κ defined respectively by Eqs. (87) and (88) in [40]. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The natural frequency of the unforced damped oscillator is therefore ωnat ¼ κ ðωnat Þ β ðωnat Þ2 : When the driving force is due to a linear-frequency-modulated wave, the Dynamic Resonance Frequency (DRF), ωDRes , at which the maximum amplitude occurs has been determined. This frequency corresponds to the first zero of the function Θr ðtÞ of the chirp-rate r defined by Eq. (45) in [41]. It is adapted here to the case of bubbles forced to oscillate at low pressure levels ðPðtÞ o 5 kPaÞ: pffiffiffi i Θr ðt Þ ¼ 2Re r ð1 þiÞϑn ðω Þ pffiffiffiffi ω ϑðω Þ (4) π h i pffiffiffiffi R u n2 2 where ϑðuÞ ¼ e u 1 þ ð2i= π Þ 0 eu du is the complex error function and ð1 þ iÞ 2 r
ω ðt Þ ¼ pffiffiffi rt þ ωinit β þ iωnat ;
(5)
ωinit is the initial frequency of the chirp and ‘*’ stands for the conjugation. As we will see in Section 4, the dynamic resonant frequency differs increasingly from the harmonic resonant frequency with the chirp rate. 2.3. Dynamics of a coupled-bubble pair For a given pair, the influence of one bubble over the other corresponds to the interaction between the incident wave, the oscillation-induced radiation and the multiple-scattering due to the mutual reflections. Based on the assumption that the interbubble spacing d does not vary with time, the coupling results therefore in an increase of the inertial (added) mass and a reduction of the damping β1 , and gives rise to a lower resonance frequency ω1 of the bubble-pair system. As far as the symmetric lowest mode is concerned, the expressions write [42] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ η þ δδr ðδ þ δr Þ2 =4 ω0 ðδ þ δr Þ ; β1 ¼ : (6) ω1 ¼ ω0 2ð1 þ η þ δδr Þ 1 þ η þ δδ r where η ¼ R0 =d. δ and δr are respectively the total and radiation damping constants. These parameters can be used to predict the dependence of the resonance frequency on the chirp rate. 2.4. Dynamics of a bubble cloud The oscillations and sound emission of a bubble cloud consisting of a large number of bubbles have been studied extensively [43–48] and it has been established that the collective oscillations of large bubble clouds give acoustic emissions as low as several hundreds of Hertz. At these low frequencies which are well below the lowest bubble resonance frequency, the effects of the bubble distribution are quite small and the speed of sound cc in the mixture therefore depends only on the void fraction α. Within the limits of moderate void fractions (several percent) and isothermal behavior, the cloud resonant frequency ωc and the damping δ have been derived [43]: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cc π u u Re Γ ωc ¼ (7) u A 2t R2 π 2 1 þ 20 A α 4
δc ¼ " A 1þ
cc R20 π 2 A2 α 4
" #
π 2 ρl ωR20
c β þ c Re Γ 4 3P gas 0 c
# (8)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where cc ¼ P 0 =ρl αð1 αÞ is the speed of sound in the mixture, the polytropic exponent is described by the complex function Γ of the driving frequency [40, Eq. (65)]. Please cite this article as: K. Czarnecki, et al., Fast bubble dynamics and sizing, Journal of Sound and Vibration (2015), http: //dx.doi.org/10.1016/j.jsv.2015.06.038i
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3. Materials and methods 3.1. Numerical simulation of bubble responses Numerical experiments were performed on an isolated bubble in a viscous fluid, using the Matlab “ODE45” function to solve the modified Rayleigh–Plesset model (1). Due to the nonlinear mixing of the pump and imaging waves, the linearly frequency-modulated US waves were observed in the side-bands apart from the imaging spectral line and the signal to noise ratios of these virtual measurements were adjusted by adding white Gaussian noise. The resonance frequency was approximated using the iterative procedure defined by Eq. (3). In order to prevent the occurrence of interactions with the pump waves as far as possible, we worked in the upper band and carried out a coherent demodulation before applying any other procedure. The pumping pressure amplitude was held constant at each measurement (P pump was set between 5 and 25 kPa), but the frequency was varied continuously and linearly from 20 kHz to 80 kHz. The imaging pressure wave applied was constant in terms of both its amplitude (P imag ¼ 10 kPa) and its frequency (f imag ¼ 1 MHz). The time-dependent changes in the radius were sampled at a frequency of 18 MHz. 3.2. Time–frequency analysis of bubble responses The behavior of the forced bubble can be characterized by taking the energy distribution of its oscillations as a function of the time and the frequency. The use of chirp-controlled waves in the Dual Frequency Method naturally leads one to perform a chirplet decomposition (quasi-matched filtering) in order to obtain a relatively high energy concentration. The envelope of the analyzing window used is described by the Blackman–Harris function: hc ðtÞ ¼ hðtÞexpðiπ rt 2 Þ;
t A ð T h =2; T h =2Þ;
(9)
where r is the chirp-rate and T h is the absolute window width. The input signal s(t) is decomposed into the chirplet transform Sðt; ωÞ as follows: Sðt 0 ; ω0 Þ ¼
TZ h =2
sðτÞhc ðτ t 0 Þexpð iω0 τÞ dτ: n
(10)
T h =2
In order to increase the signal to noise ratio in the resulting transform, the chirp rate value r chosen is equal to the slope of the instantaneous frequency of the pumping wave. To prevent the occurrence of induced cavitation in supersaturated media, low Mechanical Index (MI) conditions (MI o 0:3) are maintained. Under these conditions, the sideband relative (to imaging wave power) power density measured is between 20 and 30 dB. The time–frequency (TF) bubble response analysis involves two-steps. In the first step, the discretized TF complex spectrum S½l; k is multiplied (term by term) by a 2D mask matching the pumping wave in order to reduce most of the noise energy. The inverse chirplet transform is then applied in order to synthesize the corresponding “noise-free” signal. In the second stage, a Short-Time Fourier Transform (STFT) calculated using a narrow Blackman–Harris window (and its derivatives) gives local estimates of both the Channelized Instantaneous Frequency (CIF) and the Local Group Delay (LGD), which are directly used in the reassignment procedure to obtain a concentrated spectrogram [37–39]. A block diagram of the algorithm is presented in Fig. 1. 3.3. Experimental set-up The experiments were performed in a 2 m 3 m 0.6 m water tank in order to mitigate the potential effects of any standing waves. Microbubbles with radii ranging from 10 μm to 200 μm were generated using a hydrojet (Oral B Oxyjet 1000, Braun). A single bubble was then isolated on a 80-μm diameter wire. The advantage of using a single tethered bubble is that it prevents the occurrence of Doppler effects. The dynamics of the tethered bubble differ from those of the free bubble mainly by the dependence of the inertial (or added) mass on the contact angle [49–51]: the resonant frequency may be lowered down to 20percent. However, it was checked that in the case of a bubble with a radius ranging between 40 and 160 μm, no adhesion effects are visible in the volumetric vibrations as long as the bubble is not masked by the wire [35]. The specific measurements presented in appendix on the same tethered and then released bubbles confirm that the resonance frequency of free bubbles can be quite accurately assessed when working with bubbles attached to a wire. The size of the bubble was monitored optically using an immersed optical device comprising a 5 megapixel camera (TXG 50 Baumer) equipped with a telecentric lens having a magnification of 1.5 70 and a depth of field of 660 μm (resolution 2.3 μm per pixel). Two confocused 1-MHz frequency transducers (Imasonic, 1 MHz, f ¼ 90 mm) were used both to emit the imaging waves and to detect the corresponding echoes. To pump the bubble, a low frequency transducer was added (Ultran, GMP 50 kHz). The two transmitters were synchronized using an arbitrary waveform generator (LeCroy, ArbStudio 1104, four channels). The receiver was connected to an oscilloscope (Agilent Technologies, InfiniiVision DSO5014A, 100 MHz). The complete set-up is described in Fig. 2. A suitable pumping sweep was chosen for sizing bubbles with radii ranging between 40 μm and 160 μm corresponding to Minnaert frequencies ranging from 20 kHz to 80 kHz. The acoustic pressures at the focal point had been previously measured using low and high frequency hydrophones (Reson, Ref. TC 4034, Please cite this article as: K. Czarnecki, et al., Fast bubble dynamics and sizing, Journal of Sound and Vibration (2015), http: //dx.doi.org/10.1016/j.jsv.2015.06.038i
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Fig. 1. Two-step bubble analysis: after a noise filtering step, the dynamic resonance frequency assessment was further enhanced by applying the reassignment method.
Fig. 2. Schematic diagram of the release and sizing chain.
Precision Acoustic Ref. 1524 & DCPS 060). The maximum acoustic pressure at the focal point in the high frequency field amounted to 10 kPa and in the low frequency field to 3 kPa. 4. Results and discussion 4.1. Temporal bubble response analysis The time responses of a 64-μm bubble are illustrated in Fig. 3 in the case of linear chirp waves of various durations (100, 10 and 1 ms) at a constant pressure P p ¼ 5 kPa. When the signal duration is reduced, less energy is obviously transmitted to the bubble. Please cite this article as: K. Czarnecki, et al., Fast bubble dynamics and sizing, Journal of Sound and Vibration (2015), http: //dx.doi.org/10.1016/j.jsv.2015.06.038i
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Fig. 3. Time responses of a 64-μm bubble to short-duration up-chirp excitations (100, 10 and 1 ms). Pumping pressure level¼ 5 kPa.
Fig. 4. Time responses of a 64-μm bubble to short-duration down-chirp excitations (100, 10 and 1 ms). Pumping pressure level¼ 5 kPa.
Therefore, as shown in Fig. 3, during the swept-sine excitation, the maximum response of the resonance is lower than the maximum response of the harmonic excitation approximated here using the long 100-ms signal. In addition, as the chirp-rate increases, the ratio between the time required to reach the maximum and the signal duration also increases: this value is therefore not reached when the instantaneous frequency of the linear chirp equals the static (harmonic excitation) resonance frequency: 49 kHz in the case of a bubble with a radius of 64 μm (Fig. 3). Fig. 5 shows that the frequency shift depends not only on the chirp-rate but also on the bubble size. In practical situations, the chirp-rate is predetermined, and the Dynamic Resonant Frequency versus the bubble radius can therefore be predicted using (4). This equation can also be applied to down-chirp excitations as shown in Fig. 6 where the bubble ringdown is quantified and shows a quasi-symmetrical shape as compared with Fig. 5. Indeed, it has been established that in the absence of damping, during both run-up and run-down processes with equal absolute chirp-rates, the frequency shifts are the same [41] (Figs. 3–6). In addition, as depicted in Fig. 8, the resonance frequency decreases when the amplitude of the pumping wave increases due to the (nonlinear) softening behavior of the bubble system [52]. The frequency shift is actually more pronounced when linear down-chirps are used as shown in Fig. 9. An efficient method of solving the nonlinear inverse problem (the sizing procedure) might therefore consist in minimizing the least square error between the dynamic resonance frequency measured and that obtained using the Rayleigh–Plesset model (1). In these simulations, the amplitude of the low frequency driving chirp was kept constant across the frequency range. In more practical situations, the amplitude fluctuations in the transducer bandwidth have to be taken into account. The results presented in a previous paper [35] showed that the strong resonance ð5 o Q factor o 30Þ compensates for the amplitude fluctuations in the spanned frequency range and makes bubble sizing possible even outside the transducer 6-dB bandwidth. The frequency shift curves of a 64-μm bubble pair present the same evolution with the chirp rate as a single bubble. The reduction of inter-bubble spacing increases the coupling, which results in a lower damping (Fig. 7). Therefore, the contribution of the natural mode in the system response is relatively more important and thus the ringing-up/down effect is more pronounced than that of a single bubble. Considering the experimental conditions of [45], the frequency shift curves versus the chirp rate of a 1.5-mm bubble cloud show a similar dependence to that of a single bubble. Also we can notice a similar dependence on the respective radii. However, the damping of the cloud depends on the void fraction, which can fluctuate up to several orders of magnitude. In the case of a 5-cm (radius) cloud, with a void fraction ranging between 0.5 percent and 2 percent, the damping (the resonance frequency, respectively) ranges from 180 to 60 (from 600 to 300 Hz, respectively). This could explain the delay in the increase of the frequency Please cite this article as: K. Czarnecki, et al., Fast bubble dynamics and sizing, Journal of Sound and Vibration (2015), http: //dx.doi.org/10.1016/j.jsv.2015.06.038i
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Fig. 5. Normalized frequency shift versus the positive chirp rate with various bubble sizes (linear oscillation regime).
Fig. 6. Normalized frequency shift versus the negative chirp rate with various bubble sizes (linear oscillation regime).
Fig. 7. Normalized frequency shift of 64-μm bubble pairs versus the chirp rate for various interspacing distances.
shift versus the chirp rate in Fig. 10: a high damping factor reduces the contribution of the natural modes in the forced bubble response (see next paragraph), which results in a small perturbation of the off-resonant behavior, and therefore ωDRes =ωHRes C 1. If the chirp rate is to be high however, the resonant phenomenon prevails, inducing a significant frequency shift.
4.2. Time–frequency bubble response analysis Time–frequency distributions can be advantageously used to size microbubbles when the low-chirp rate (r¼0.6 MHz/s) DUFM is used, especially in the presence of noise. In comparison with standard approaches based on the classical spectrogram, the reassignment procedure improves the resolution and decreases the probability of false bubble detection, as shown by the spectral marginals presented in Figs. 11 and 12. These techniques were applied to the experimental characterization of microbubbles tethered to a 80-μm wire (Fig. 13). The results presented in Fig. 14 in the case of a 10-ms signal (r ¼6 MHz/s) show that a clearly visible pattern of energy Please cite this article as: K. Czarnecki, et al., Fast bubble dynamics and sizing, Journal of Sound and Vibration (2015), http: //dx.doi.org/10.1016/j.jsv.2015.06.038i
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Fig. 8. Time responses of a 64-μm bubble to short-duration up-chirp (100, 10 and 1 ms) at a pressure level of 25 kPa.
Fig. 9. Time responses of a 64-μm bubble to short-duration down-chirp (100, 10 and 1 ms) at a pressure level of 25 kPa.
distribution was obtained in the TF domain under laboratory conditions, giving an estimated radius of 38 μm. The bubbles were sized by calculating the resonance frequency from the (classical and concentrated) spectrograms. However, as shown in Fig. 15, short-duration bubble sizing (with a duration of 2 ms) when applying spectrogram techniques results in strong blurring effects, which affects the precision of the results, whereas the concentrated spectrogram calculated from the noise-free signals re-synthesized re-allocates the blurred energy to a coherent location, as can be seen from Fig. 16. This figure shows the dynamic resonance of a 79-μm microbubble excited with a 2-ms signal with relatively high sharpness. Even in the case of high chirp rates (of the order of 30 MHz/s), the corrected DFUM will therefore give relatively reliable detection and characterization performances. On the other hand, the bubble radius estimated using the maximum amplitude approach would have been 73 μm, corresponding to an underestimate of more than 7.5 percent of the equilibrium radius. In addition, the concentrated spectrogram depicted in Fig. 16 shows that the linearity of the frequency modulation (which exhibits a visible bend) is altered at the resonance frequency. Oscillations with a constant frequency superimposed on the volumetric pulsations forced by the pumping chirp persisted for about one-tenth of a millisecond. In the time– frequency domain, the concentrated spectrogram therefore shows a typical Y-shaped pattern which was also clearly obtained in the simulation illustrated in Fig. 17. This bubble response can be interpreted as the superimposition of the free oscillations at the natural frequency (i.e., the impulse response of the bubble), triggered during the passage of resonance, and forced oscillations with an increasing excitation frequency. In practice, this specific pattern should improve the bubble detection and characterization using pattern recognition methods.
5. Conclusion The aim of the present study was to develop short-duration bubble detectors and sizers based on the Dual Frequency Ultrasound Method (DFUM). A fast single bubble characterization was obtained with this method using relatively large chirp-rate pumping waves. These waves are required when short-duration (corresponding to a duration of a millisecond) and/or high repetition rate inspection is performed on moving media such as blood. This method can be used to detect and characterize circulating bubbles in the right ventricle for the prevention of decompression sickness, for example. Please cite this article as: K. Czarnecki, et al., Fast bubble dynamics and sizing, Journal of Sound and Vibration (2015), http: //dx.doi.org/10.1016/j.jsv.2015.06.038i
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Fig. 10. Normalized frequency shift of spherical clouds (radius 5 cm) of 1.5-mm bubbles versus the chirp-rate with various void fractions
Fig. 11. Classical spectrogram of the simulated DFUM signal backscattered by a 64μm bubble and the marginal distributions. The pumping linear chirp frequencies were 20–80 kHz, the duration was 100 ms, and the imaging wave had a central frequency of 1 MHz.
Fig. 12. Concentrated chirplet spectrogram obtained after reassigning the (pre-)processed data. Same simulated conditions as in Fig. 11.
Fig. 13. Spectrogram obtained from an experimentally measured bubble (estimated radius: 38 μm) (with a 10-ms chirp).
Numerical simulations based on a modified Rayleigh–Plesset model have shown that chirp-wave activation leads to a time shift in the maximum amplitude of the bubble response. By analogy with harmonic activation, the dynamic resonance frequency is that of the component which induces the largest oscillations. In the (quasi-)linear regime, the resonant frequency shift, which was found to depend on both the chirp rate and the bubble radius, was quantified numerically.
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Fig. 14. Concentrated spectrogram obtained from an experimentally measured bubble (estimated radius: 38 μm) (with a 10-ms chirp).
Fig. 15. Spectrogram of a 2-ms experimental bubble response. The radius was estimated at 79 μm.
Fig. 16. Same bubble response as in Fig. 15. Reassignment method giving a bubble-radius estimate of 79 μm.
Fig. 17. Concentrated spectrogram of a simulated 64-μm bubble response to a 2-ms chirp wave at a pressure level of 5 kPa. The bubble resonance was characterized by a “Y” shaped pattern of the energy distribution. The harmonic resonant frequency (49 kHz) is clearly visible on the marginal spectrum.
A dedicated TF estimator was also designed to calculate the concentrated spectrogram (based on the reassignment technique) of the noise-free signal obtained using a matched-filtering module. Using this estimator, tethered bubble were experimentally characterized with durations ranging from 10-ms down to 2-ms: the radii of the bubbles detected were in the 40–160 μm range. Under laboratory conditions, the sizing obtained was qualitatively much more accurate than the usual procedure based on the standard spectrogram. Please cite this article as: K. Czarnecki, et al., Fast bubble dynamics and sizing, Journal of Sound and Vibration (2015), http: //dx.doi.org/10.1016/j.jsv.2015.06.038i
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Fig. 18. Follow up of three consecutive images (every 66 ms) of a jumping bubble activated by a 30-kPa 100-cycles burst of frequency equals to the resonance frequency (45 kHz) of the bubble. The bubble is momently pushed away by the radiation force and then attracted to the wire by the secondary Bjerknes forces.
Fig. 19. Image sequence (frame rate 15 Hz) describing the evolution of a 60-μm bubble during the ultrasonic release (below A) and sizing (B) process.
Fig. 20. Comparison between the resonance amplitude modulation curves before (a) and after (b) the detachment of the bubble. The corrected (compensation for the frequency shift) values obtained were 58.9 and 59:9 μm, respectively, in good agreement with the optical measurements (60 μm). The 1 μm difference shows that tethered bubbles are representative of the free-bubble dynamics.
In addition, the concentrated spectrogram makes it possible to keep a good localization in the TF plane, although large up-chirp rates (of up to 30 MHz/s) are used. In the range of bubble radii covered, the sharpness of the concentrated spectrogram made it possible to observe that when millisecond activations are employed, the free response (i.e., the impulse response) of the bubble is excited in addition to the forced response. The observed frequency shift results from the superimposition of these two responses; neglecting this dynamic effect may lead to under-estimating the bubble size by up to 20 percent. The same conclusion applied to pairs and clouds of coupled bubbles.
Acknowledgment This work was partly supported by the French Research Ministry under the “Smart US” project, Ref. ANR-10-BLAN-0311 and MUSEM-2015-IC-STAR project, by the Provence-Alpes-Cte-D'Azur Council, by the Cancrople PACA and by the French Please cite this article as: K. Czarnecki, et al., Fast bubble dynamics and sizing, Journal of Sound and Vibration (2015), http: //dx.doi.org/10.1016/j.jsv.2015.06.038i
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Ministry of Defense, under the “BORA” project. Authors are grateful to Eric Debieu for his contribution in the experimental set up.
Appendix A. Effects of the tethering on the resonance The question of the possible effects of the adhesion to a rigid boundary on the resonance has been previously addressed in [49,50]. It was established that the resonance frequency shows an angle-contact dependence and can be altered by up to 18 percent. However, in our experiments, with bubble radii ranging between 40 and 160 μm, no adhesion effects were visible as long as the bubble was not masked by the 80-μm diameter copper wire. In order to determine the effects of the tethering, measurements were performed on a bubble attached to the wire, and on the same bubble after it had been released and rose due to buoyancy. For this purpose, an ultrasound-driven release process was designed: this process is presented below. Due to surface tension, a bubble in contact with a surface is subject to capillary forces which keep it attached. When a focused high frequency (above the resonance frequency) ultrasound wave is impinging on the bubble, the radiation force pushes the bubble away from its equilibrium position. In addition, because of the pressure gradient, primary Bjerknes forces will tend to move the bubble away from the focus, eventually towards a pressure node. In addition to these forces, secondary Bjerknes forces are induced by the oscillations of the bubble in close proximity to the rigid (and possibly vibrating) wire surface. If the wave frequency chosen is near the resonance frequency, both of the Bjerknes forces will be maximum. Lastly, the buoyancy also acts on the bubble. When an ultrasonic wave hits it, the bubble is pushed away before it enters a steady-state oscillation regime. Once the oscillations have started, they give rise to additional attractive forces towards the wire. The duration of the setting-up of the attractive force is of the order of Q-factor (quality factor) cycles. In the case of a 80-μm bubble, the Q-factor is about 10. If a 100-kPa ultrasonic wave with a central frequency 40 kHz acts on a 80-μm bubble (Minnaert frequency of 40 kHz), the primary and secondary Bjerknes forces will amount to 10 6 N and 10 2 N respectively, the radiation force to 10 5 N and the buoyancy force to 10 8 N. The capillary force cannot be reliably estimated since the contact angle is not visible. This rough evaluation shows that secondary Bjerknes forces are four orders greater than the other forces and need to be minimized. Therefore, 10-cycle pump bursts are used to kick off the bubble without inducing any significant attractive forces. Fig. 18 illustrates how a 100 cycle burst can be used to detach the (jumping) bubble from the wire, and how it is also responsible for its re-attachment. The image sequence (frame rate 15 Hz) presented in Fig. 19 shows the evolution of a 60-μm (optically measured) bubble during the complete sizing process. Once the bubble has been released, the vertical rising trajectory is suddenly shifted to the right at the instant when the pump wave is applied again for the sizing (the transducer was located here on the left side of the image, delay 200 ms). The resonance frequencies can thus be estimated first with the bubble attached (Fig. 20a) and secondly, on the free bubble (Fig. 20b). For this purpose, ten 10-ms chirps are transmitted (chrip-rate¼3 MHz/s), the back-scattering image waves are processed and the amplitude modulation curves are averaged. The corresponding maximum radii amount to 58 μm and 57 μm, respectively. A higher signal to noise ratio can be observed on the amplitude modulation curves of the free bubble, since the focus of the transducer was located several millimeters above the static position of the bubble in order to obtain optimum free-bubble measurements. The relative frequency shift associated with a 60-μm bubble with a 3-MHz/s chirp rate amounts to 1.033. The corrected resonant radii estimated were therefore 58.9 μm and 59.9 μm. These values are in very good agreement with the optical measurements. Reality is certainly rather more complex and we expect greater stiffness and damping to exist due to the adhesion. These parameters could be estimated using greater optical and temporal resolutions. However, in a first approximation, the tethered-bubble dynamics described here seem to be a representative approximation of free bubbles behavior. References [1] T.G. Leighton, From seas to surgeries, from babbling brooks to baby scans: the acoustics of gas bubbles in liquids, International Journal of Modern Physics B 18 (25) (2004) 3267–3314. [2] A.D. Phelps, D.G. Ramble, T.G. Leighton, The use of a combination frequency technique to measure the surf zone bubble population, Journal of the Acoustical Society of America 101 (4) (1997) 1981–1989. [3] H. 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Please cite this article as: K. Czarnecki, et al., Fast bubble dynamics and sizing, Journal of Sound and Vibration (2015), http: //dx.doi.org/10.1016/j.jsv.2015.06.038i