Study of non-spherical bubble oscillations near a surface in a weak acoustic standing wave field Xiaoyu Xi and Frederic Ceglaa) Department of Mechanical Engineering, Imperial College London, London SW7 2AZ, United Kingdom
Robert Mettin Christian Doppler Laboratory for Cavitation and Micro-Erosion, Drittes Physikalisches Institut, Georg-August-Universit€ at G€ ottingen, Friedrich-Hund-Platz 1, 37077 G€ ottingen, Germany
Frank Holsteyns and Alexander Lippert Lam Research AG, SEZ-Strasse 1, 9050 Villach, Austria
(Received 10 April 2013; revised 13 January 2014; accepted 22 January 2014) The interaction of acoustically driven bubbles with a wall is important in many applications of ultrasound and cavitation, as the close boundary can severely alter the bubble dynamics. In this paper, the non-spherical surface oscillations of bubbles near a surface in a weak acoustic standing wave field are investigated experimentally and numerically. The translation, the volume, and surface mode oscillations of bubbles near a flat glass surface were observed by a high speed camera in a standing wave cell at 46.8 kHz. The model approach is based on a modified Keller-Miksis equation coupled to surface mode amplitude equations in the first order, and to the translation equations. Modifications are introduced due to the adjacent wall. It was found that a bubble’s oscillation mode can change in the presence of the wall, as compared to the bubble in the bulk liquid. In particular, the wall shifts the instability pressure thresholds to smaller driving frequencies for fixed bubble equilibrium radii, or to smaller equilibrium radii for fixed excitation frequency. This can destabilize otherwise spherical bubbles, or stabilize bubbles undergoing surface oscillations in the bulk. The bubble dynamics observed in experiment demonstrated the same trend as the theoretical results. C 2014 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4864461] V PACS number(s): 43.25.Yw, 43.20.Ks, 43.35.Yb, 43.25.Qp [TGL]
I. INTRODUCTION
Understanding the interaction of acoustically driven bubbles and nearby solid or flexible boundaries is essential in many applications of ultrasound and acoustic cavitation. The most prominent effects of strong bubble oscillations at solids are material erosion1 and cleaning of attached contaminations.2 Both are typically associated with high pressure peaks and strong shear flows occurring during bubble collapse3 and possible liquid jet impact.4,5 Other observed physical and chemical effects, like surface activation or enhanced chemical reactions of solids in liquids,6 are due to extremes in pressure or temperature caused by close or attached bubbles during collapse. The strong shear flow associated with heavy collapse and jetting also causes substantial forces on flexible boundaries, where tissue and cell membranes are of particular interest.7,8 A typical and standard application is the disintegration of cells by ultrasonic cavitation.9 On the other hand, the flow field of only weakly or moderately oscillating acoustically driven bubbles has proven to be effective in more delicate applications like sonoporation10 or removal of small particles from structured substrates.11,12 Here, pressure shocks and fast jet impact can be counterproductive or harmful, and should be avoided. Nevertheless, significant perturbations of substrate or membrane boundary a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]
J. Acoust. Soc. Am. 135 (4), April 2014
Pages: 1731–1741
layers seem achievable, and by controlling the bubble oscillations and the mediated shear at the boundaries, driven bubbles might be engineered as a tool.13 Thus, it is necessary to consider in detail the influence of the boundary on weak to moderate bubble oscillations and the associated flow field surrounding the bubbles. While existing literature is related to this topic, there are still many open questions. In this work, we address the question of the onset of non-spherical mode oscillations of driven single bubbles directed toward and reaching a solid surface which is investigated experimentally and compared to theory. Assuming a bubble remains spherical to first order, a neighboring wall could force the bubble to change its behavior when it is oscillating on the surface. Suslov et al.14 calculated the nonlinear response of micro-bubbles near a rigid wall based on a modified Keller-Miksis equation. The resonance frequency of a bubble that oscillates on a wall was simulated by Strasberg15 and Payne et al.16 It has been recognized that the resonance frequency of a bubble near a rigid wall is lower than that of a free bubble. However, Doinikov et al. found out that the elasticity of a wall cannot be neglected. They proposed several modified Rayleigh-Plesset equations to explain the bubble dynamics on elastic walls of different properties.17–19 Experimental work was also carried out in the past to understand the spherical bubble behavior near a wall. Garbin et al.20 observed the change of bubble dynamics near a boundary using a high speed camera system.
0001-4966/2014/135(4)/1731/11/$30.00
C 2014 Acoustical Society of America V
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Besides the spherical oscillation, a bubble can also oscillate non-spherically in an ultrasound field. In a bulk medium, the spherical shape stability was first modeled by Plesset.21 Later on, the viscous effect in the vicinity of a bubble surface was considered by Prosperetti and Hao.22 By only taking the local vorticity into account, Hilgenfeldt et al.23 studied the shape instabilities and diffusive instabilities for a single sonoluminescence bubble. Loughran et al.24 considered the shell properties of Ultrasound Contrast Agents (UCAs), and calculated the non-spherical behavior of shelled micro-bubbles. Maksimov and Leighton25 developed a model that explains the selection of the regular pattern on the bubble wall. In contrast, the influence of a wall on the nonspherical bubble oscillation is still not completely clear. Also, due to the complicated bubble-wall interaction, little experimental work on non-spherical bubble oscillation can be found in the literature. Recently, the nonspherical behaviors of UCAs within micro-vessels were reported by Zhao et al.26 and Caskey et al.,27 respectively. Vos et al.28 measured the non-spherical shapes of UCAs near cellulose walls, and analyzed the corresponding shear stress exerted on the boundary. Versluis et al.29 observed the shape oscillations of micro-bubbles excited through parametric driving without considering the wall effects. In these previous experimental studies, non-spherical bubbles were normally placed at the monitoring positions prior to the observations. In many real life applications, bubbles need to be transported to a target area because they are generated far from the surface. It is well-known that a bubble’s non-spherical oscillation modes interact with each other during its translation. In a bulk medium, this coupling effect between different modes was theoretically investigated by Longuet-Higgins30 and Doinikov.31 Nevertheless, little is understood about the translation of a non-spherical bubble near a wall. The change of nonspherical oscillation modes before and after the bubble’s arrival on a boundary has not been investigated. This is due to the lack of a reliable experimental configuration that can control the bubble oscillation modes in the vicinity of a boundary. In this paper, we report the influence of the translation toward a neighboring wall on the nonspherical bubble oscillations. In the experiments, the bubble motion control was achieved using a multi-layered stack that was designed based on a one-dimensional matrix transducer model.32,33 The test rig has been verified as an effective tool to control single and multi-bubble transportation in an acoustic standing wave field.34,35 Numerically, the dynamics of non-spherical bubbles near a wall are analyzed based on modified Keller-Miksis equations, a dynamical equation, and a pair of translation equations. This paper is organized as follows: Section II shows the experimental setup. In Sec. III, the theoretical background of non-spherical bubble dynamics near a surface is presented. The observed bubble oscillation modes are demonstrated in Sec. IV, and are discussed in Sec. V. Conclusions are drawn in Sec. VI. 1732
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II. EXPERIMENTAL CONFIGURATION
A schematic diagram of the experimental setup is shown in Fig. 1. Bubbles of radii ranging from 10 to 80 lm were generated by an electrolysis method and contained hydrogen. The bubble motion was manipulated within a multi-layered stack that consists of an ultrasonic transducer and a water chamber with side windows. The acoustic field in the water corresponds well to a one-dimensional standing wave in the x direction (Fig. 1). The transducer was operated at a driving frequency of 46.8 kHz. At this frequency, the sizes of the test bubbles are much smaller than the acoustic wavelength. A high speed camera system (FastCam SA5, Photron, San Diego, CA) was used to visualize the bubble dynamics near a surface at various frame rates up to 525 000 frames/s. A flat target glass plate of 0.1 mm thickness (VWR, United Kingdom) was inserted parallel to the wave front at x ¼ 4 mm (the origin of the coordinate system is selected at the transducer-water boundary, x ¼ 0, y ¼ 0, z ¼ 0). The water chamber was terminated by a similar glass plate against air at x ¼ 8 mm. More details of the test rig were reported in previous work.34,35
III. THEORY
When the acoustic pressure amplitude reaches a certain value, a driven bubble can start to oscillate non-spherically in the sound field. It is well-known that the non-spherical modes are related to their spherical counterpart. The spherical oscillation can be modeled by different radial oscillation equations depending on the bubble-wall separation distance. In this section, two cases of non-spherical oscillations are considered: A bubble translating toward a wall, and a bubble oscillating on a boundary. To calculate the non-spherical modes, let us first introduce an initial small perturbation on a bubble’s surface, which is given by
FIG. 1. (Color online) A schematic diagram of the test rig for studying the bubble behavior. The high speed camera system is used to focus on the field of view at 525 000 frames/s. The bubble is transported from the injection point (x ¼ 5 mm) to glass 1 (x ¼ 4 mm) within an acoustic standing wave field. The predicted pressure profile in the water layer is shown. Xi et al.: Study of non-spherical bubble oscillations
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Rðh; tÞ ¼ RðtÞ þ an ðtÞYn ðhÞ;
(1) x0free
where R(t) is the time-varying bubble radius, Yn(h) is the spherical harmonic of the nth order with amplitude of an(t). Including the local viscous effect, the distortion amplitude an(t) takes the form of23 a€n ðtÞ þ Bn ðtÞa_ n ðtÞ An ðtÞan ðtÞ ¼ 0;
d ðn 1Þðn þ 2Þ þ 2nðn þ 2Þðn 1Þ ; R _ 3R 2 d þ 2 ðn þ 2Þð2n þ 1Þ 2nðn þ 2Þ2 ; Bn ðtÞ ¼ R R R
(2)
where r is the surface tension, q is the liquid density, is the kinematic viscosity, x is the angular frequency, and bn ¼ (n 1)(n þ 1)(n þ 2). The mode coupling between the mode n ¼ 0 (radial) to higher modes n ¼ 2,3,…, etc., and the mode n ¼ 1 to n ¼ 0 is considered. The coupling between higher modes n ¼ 2,3,…, etc., is neglected. To excite a non-spherical oscillation, the external pressure needs to exceed a certain threshold Pthreshold,36,37
Pthreshold
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u ðs 1Þ2 þ 4p h ¼ qR20 u ; 2 u t 3 1 þ q2 s þ 2p þ 2 l þ 2 2
h ¼ ðx20 x2 Þ2 þ ð2btot xÞ2 ; 4ðn 1Þðn þ 1Þ þ ðn þ 2Þr ; qx2 R30 " #2 2ðn þ 2Þð2n þ 1Þ p¼ ; xR20 s¼
q¼
6ðn þ 2Þ ; xR20
(4)
where c is the polytropic exponent of the gas inside the bubble, and Pv is the vapor pressure in the liquid. Previous studies showed that the resonance frequency x0free is shifted downward when the bubble oscillates on a rigid wall. The lowered resonance frequency is given by16 (5)
where dwall is the separation distance between a bubble and the wall. It is worth mentioning that Payne et al.16 only considered the monopole component in analyzing the interaction between the original bubble and its mirror counterpart. However, when the separation distance is comparable to the bubble size, the multipole expansion terms need to be taken into account for a more accurate solution. Moreover, when the bubble size is smaller than the wavelength, the pressure inside the bubble is constant because the equilibrium pressure P0 is larger than that from the surface tension. The bubble surface can be treated as equipotential. This analogy with electrostatics was discussed by Kobelev and Ostrovskii.39 A similar method was also used by Oguz and Prosperetti40 for studying the bubble dynamics near a free surface. Since the theoretical predictions are only used here to describe the qualitative trend of bubble behavior and the use of Payne’s result will not cause severe errors as seen in the following experimental results, the investigation of an accurate analytical solution of the flow surround the bubble will not be discussed. For a bubble far away from the wall (dwall R0), the influence of the wall is negligible (x0 ¼ x0free). On the other hand, for a bubble that is attached on a rigid wall, Eq. (5) indicates that x0 is lower pffiffiffiffiffiffiffithan the free bubble resonance frequency x0 ¼ x0free = 1:5 which is not very different from the solutionpffiffiffiffiffiffiffiffiffi obtained by Kobelev and Ostrovskii39 x0 ¼ x0free log2 . The damping effect of a bubble has been extensively studied in the literaturep(see Ref. 38). A typical length scale ffiffiffiffiffiffiffiffiffiffiffi for viscous effect d ¼ 2=x is introduced here. When the length scale is relatively thin, the damping factor for the non-spherical modes is25
(3)
where R0 is the equilibrium bubble radius which is much smaller than the acoustic wavelength (R0 k). btot is the total damping factor and x0 is the angular resonance frequency. It is worth mentioning that Eq. (3) is related to two resonance factors (x0 and btot). The presence of a nearby wall could introduce correction terms into these two factors. Therefore, even small changes in x0 and btot can result in a significant difference in the prediction of pressure threshold. The angular frequency x0 has been thoroughly examined in the literature (see Ref. 38, for example). For an un-coated bubble, x0free is given by J. Acoust. Soc. Am., Vol. 135, No. 4, April 2014
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2r 2r 3c P0 þ Pv þ Pv ; R0 R0
x0free x0 ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; R0 1þ 2dwall
R€ b r 2 R_ An ðtÞ ¼ ðn 1Þ n 3 3 R R qR
rffiffiffiffi ! R d ¼ min ; x 2n
1 ¼ pffiffiffi R0 q
bn ¼ ðn þ 2Þð2n þ 1Þ
: R20
(6)
The total damping factor is a combined effect of viscous, thermal, and acoustic damping. The damping factor for the breathing mode n ¼ 0 is estimated by a linear analysis as25 rffiffiffiffiffiffi x2 R0 2 x0wall D þ 2 þ 3ðc 1Þ b0 ; (7) 2c 2R0 2x R0 where D is the diffusivity coefficient. When a bubble oscillates near a wall, the boundary can be replaced by a mirror bubble on the other side of the wall. Xi et al.: Study of non-spherical bubble oscillations
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The mirror bubble oscillates in phase with the original one at the same amplitude in the presence of a rigid wall.41 As seen in Eq. (2), the amplitude of the original bubble’s nonspherical oscillation is closely related to its spherical radius R(t). R(t) can be obtained from the two radial oscillation equations depending on the bubble-wall separation distance (dwall). Figures 2(a) and 2(b) show the cases when a bubble translates toward an elastic boundary and when it oscillates on the surface, respectively. In both cases, dwall is assumed to be larger than R0. The bubble radial oscillation equation is welldocumented in the literature. Recently, Doinikov et al.17,19 pointed out that the oscillation of a bubble near an elastic wall depends on the separation distance (dwall), wall density (qwall), and thickness (Lwall). Donikov et al. neglected the shear strain of the wall because the motion of a soft wall is mainly caused by the compressive liquid movement. In the present case, however, such a simplification may not be valid because the shear wave velocity of glass is higher than soft tissues. Hence, the normal flow velocity on the glass surface is almost zero which means the glass plate can be treated as a rigid wall. The radial oscillation of a bubble near a rigid wall is governed by a modified Keller-Miksis equation.14,34,35 The radial equation proposed by Xi et al.35 considers the collective behavior of a group of bubbles. In the model, the bubble-wall interaction is represented by a bubble-imaginary-bubble system. The modified Keller-Miksis equation is hence given by 3 R_ _ 2 R_ € 1 RR þ R 2 2c c x_ 2 1 R 1 R_ 2 € 1þ ¼ ð2R_ R þ R2 RÞ; Psc þ P_ sc þ q qc 4 2dwall c c 2r R30 a3 2r 4gR_ P0 Pex ; Psc ¼ P0 þ 3 3 R a R0 R R Pex ¼ Pa sinðxtÞsinðkdnode Þ;
(8)
where c is the sound velocity in the liquid, and Psc is the scattered pressure. The bubble center position is given by (x, y) in a two-dimensional system. P0 is the hydrostatic pressure, a is the van der Waals hard core, and Pex is the external driving pressure which takes the form of a standing wave. Pa is the pressure amplitude at the antinode, k is the wave number, and dnode is the separation distance between a bubble center and a pressure node. The second term on the right-hand side of Eq. (8) ðx_ 2 =4Þ represents the coupling between the translation and radial oscillation, while the last term is the contribution of the imaginary bubble on the radial oscillation of the original one. The translation equations governing the bubble motion in the x and y axes are34,35,42,43 3R_ x_ 3Fx ¼ ; 2pqR3 R 3Fy 3R_ y_ ¼ ; y€ þ 2pqR3 R x€ þ
(9)
where Fx and Fy are external forces in the x and y axes, respectively. Figure 2(b) shows another case when a bubble oscillates on a boundary. In this case, the bubble surface displacement is substantially influenced by the wall. By removing the translation term in the modified Keller-Miksis equation, the radial equation for a bubble on the glass surface is 3 R_ _ 2 R_ € RR þ 1 R 2 2c c 1 1 R_ R d 2 € 1þ þ ¼ ð2R_ R þ R2 RÞ: Psc q 2dwall c c dt (10) The amplitude of non-spherical modes an can be solved by combining Eqs. (2), (8), and (10) together. It needs to be pointed out that non-spherical oscillation is a complicated
FIG. 2. The spherical oscillation of a bubble (a) at a separation distance dwall from an elastic boundary, and (b) on the boundary. A mirror bubble is introduced to replace the boundary. This mirror bubble oscillates in phase with the original bubble near a rigid wall.
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Xi et al.: Study of non-spherical bubble oscillations
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process, and the solved an is only an approximation of the actual non-spherical behavior. Nevertheless, the aim of this study is to qualitatively understand the changes of bubble non-spherical modes near a wall. As seen later in this paper, the above method provides a qualitative insight into the bubble non-spherical modes. The development of an improved non-spherical model is planned in the future. IV. RESULTS
The values of the parameters used in this study are f ¼ 46.8 kHz, q ¼ 1000 kg/m3, qwall ¼ 2230 kg/m3, P0 ¼ 101.3 kPa, c ¼ 1480 m/s, r ¼ 0.072 N/m, c ¼ 1, g ¼ 0.001 Pa*s. The dimensions of objects in the videos were calibrated with a standard 300 lm wide stick. The pressure amplitude was calculated based on a onedimensional matrix transducer model, and was verified experimentally.32,33 As seen in Fig. 1, the sound pressure amplitude monotonically decreases from x ¼ 0 mm at the water-transducer interface to x ¼ 8 mm at the water-glass contact surface. In Secs. IV A and IV B, the pressure amplitudes that are discussed refer to the pressure on the righthand side surface of glass 1 (x ¼ 4.1 mm). A. Spherical bubble oscillation near a surface
At 6.9 kPa, a bubble of 43 lm mean radius translates toward glass 1 as shown in Fig. 3(a). The bubble maintains a spherical shape at this pressure amplitude and its radius periodically varies between 41 and 45 lm. The separation distance between the bubble and the boundary is 0.3 mm. Since the driving frequency is set as 46.8 kHz, the time of one acoustic cycle is 21 ls. The three frames in Fig. 3(a) represent the bubble oscillation over about one acoustic cycle. After 11 ms, the bubble arrives onto the surface and still keeps its spherical shape [Fig. 3(b)]. The maximum radius, however, increases from 45 to 50 lm, while the minimum radius is lowered from 41 to 39 lm. It is worth mentioning here that a nearby wall can distort a bubble’s surface when the bubble arrives on the wall. The spherically symmetric flow field around a free bubble cannot be established due to the boundary. At pressures below the threshold for parametric excitation of surface waves, the bubble volumetric oscillation is not spherically symmetric. However, the asymmetric shape is not visible from the
FIG. 3. Selected frames from a video of bubble oscillation near a surface at 6.9 kPa. The bubble mean radius is 43 lm. (a) The bubble oscillates with a spherical mode when approaching glass 1. The three frames represent the bubble oscillation over one acoustic cycle. (b) The bubble keeps its spherical shape when sitting on the glass surface. The bubble is supposed to touch glass 1 at a contact point. The scale bar represents 200 lm. The time of one acoustic cycle is 21 ls. J. Acoust. Soc. Am., Vol. 135, No. 4, April 2014
experimental results because it is small (proportional to the amplitude of the radial oscillations). When the pressure is above the instability threshold, surface waves are stronger than the volumetric oscillation result, and the surface waves are clearly seen in the photographs. At the same pressure amplitude, for a similar sized bubble, its spherical oscillation can be switched to a nonspherical one after merging with a neighboring bubble. In Fig. 4(a), a bubble (radius of 41 lm) oscillates spherically during its translation to glass 1. The spherical oscillation is disturbed by the arrival of another nearby bubble (radius of 28 lm) at 88 ms. The coalescence of the two bubbles is clearly seen in Fig. 4(b). The newly formed bubble (radius of 60 lm) then starts to oscillate with a non-spherical mode on the surface as shown in Fig. 4(c). At a higher pressure amplitude, 12 kPa, a bubble of 43 lm radius oscillates with a spherical shape where the bubble radius varies between 38 and 48 lm [Fig. 5(a)]. When the bubble arrives onto the surface in Fig. 5(b), it starts to lose its symmetric shape, and gradually shifts to a nonspherical oscillation mode. The non-spherical shape is finally formed in Fig. 5(c). The bubble starts to stretch in the horizontal direction where it reaches a maximum extension and then shrinks. The compression of the surface shape in the horizontal direction results in a growth of bubble in the vertical direction. When the bubble surface grows to the maximum in the vertical direction, its shape falls back to the original state and then repeats the whole process again. The influence of the wall on the oscillation of a bubble translating with a spherical shape is more substantial at 17.2 kPa. Initially, a spherical bubble translates in the sound field as seen in Fig. 6(a). After 1.6 ms, the bubble approaches glass 1 at a faster speed. Immediately, the bubble starts to change its shape from spherical to elliptical even without touching the glass surface. The shape change becomes more dramatic after the arrival of the bubble on the glass surface. The elongation of the bubble shape in the vertical direction eventually leads to a split of the bubble into two parts which
FIG. 4. Selected frames from a video of bubble oscillation near a surface at 6.9 kPa. The initial bubble radius is 41 lm. (a) The bubble oscillates without non-spherical modes when it is approaching glass 1. (b) A second bubble approaches the target and merges with the first one. (c) The newly formed bubble of 60 lm radius oscillates with a non-spherical shape. (Scale bar: 200 lm, the period of one acoustic cycle: 21 ls.) Xi et al.: Study of non-spherical bubble oscillations
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FIG. 5. Selected frames from a video of bubble oscillation near a surface at 12 kPa. The initial bubble radius is 43 lm. (a) The bubble oscillates spherically when it is approaching glass 1. (b) The bubble arrives on the surface (c) The bubble oscillates with a non-spherical shape of mode ¼ 3. (Scale bar: 200 lm, the period of one acoustic cycle: 21 ls.)
then merge back together to form a new bubble. Two subsequent coalescences of the target bubble with two other neighboring bubbles are also shown in Figs. 6(c) and 6(d), respectively. The newly formed bubble oscillates with a much more complicated mode and completely loses its spherical shape. B. Non-spherical bubble oscillation near a surface
On the other hand, a bubble can translate with a nonspherical shape when its size exceeds a certain threshold. At 6.9 kPa, for example, a bubble of 54 lm radius oscillates non-spherically when it moves toward glass 1 [Fig. 7(a)]. The triangle shape indicates that the bubble pulsates at the third mode during its translation. This triangle shape shifts back to a spherical one after the bubble’s arrival on the glass surface in Fig. 7(b). The temporary spherical shape, however, starts to grow back to a non-spherical one after a few acoustic cycles. The asymmetric surface shape is finally formed in Fig. 7(c). At 12 kPa, a bubble of 52 lm radius also shows a nonspherical oscillation behavior when it travels toward glass 1.
FIG. 6. Selected frames from a video of bubble oscillation near a surface at 17.2 kPa. The initial bubble radius is 43 lm. (a) The bubble oscillates with a spherical shape when it is approaching glass 1. (b) The bubble arrives on the surface. (c) Coalescence of the bubble with a nearby bubble. (d) A second coalescence of two bubbles on the surface. (Scale bar: 200 lm, the period of one acoustic cycle: 21 ls.) 1736
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FIG. 7. Selected frames from a video of bubble oscillation near a surface at 6.9 kPa. The initial bubble radius is 54 lm. (a) The bubble oscillates with a non-spherical shape (n ¼ 3) when it is approaching glass 1. (b) The bubble shape returns to a spherical one after its arrival on the glass surface. (c) The bubble oscillates with a non-spherical shape again. (Scale bar: 200 lm, the period of one acoustic cycle: 21 ls.) Xi et al.: Study of non-spherical bubble oscillations
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A similar triangle shape is seen in Fig. 8(a). Unlike the low pressure case, the bubble surface immediately changes to a more complicated shape after its arrival on the glass surface. This non-spherical shape keeps evolving throughout the following time as shown in Figs. 8(b) and 8(c). The identification of the non-spherical mode, however, is difficult to be given here owing to the limited resolution of the frames. Similarly, a bubble of 56 lm radius experiences a strong oscillation at 17.2 kPa in Fig. 9. An irregular surface shape is seen for this bubble during its translation. Glass 1 forces the bubble to change its shape even before it is hitting on the surface. The non-spherical shape becomes more substantial when the bubble starts to oscillate on the surface [Figs. 9(b) and 9(c)]. V. DISCUSSION OF THE EXPERIMENTAL RESULTS
As seen in Sec. IV, a bubble can experience various oscillation modes under different conditions. In this section, the observed non-spherical bubble oscillation mode is explained by the model shown in Sec. III. The influence of external pressure amplitude, bubble size, boundary condition, and neighboring bubbles on the bubble oscillation mode is discussed. A. Mode number identification
The images shown in Sec. IV demonstrate the various bubble oscillation modes near a wall under different external conditions. A bubble oscillating with higher modes is a complicated three-dimensional (3D) object. It is possible that the higher surface modes can project different images on different observing planes. Ideally, the reconstruction of the bubble surface modes requires the employment of several high
FIG. 8. Selected frames from a video of bubble oscillation near a surface at 12 kPa. The bubble initial radius is 54 lm. (a) The bubble oscillates with a non-spherical shape when it is approaching glass 1. (b) The arrival of the bubble on the surface. (c) The bubble oscillates with a non-spherical shape on the surface. (Scale bar: 200 lm, the period of one acoustic cycle: 21 ls.) J. Acoust. Soc. Am., Vol. 135, No. 4, April 2014
speed cameras that are located at different locations in order to capture the surface changes from all possible angles. Admittedly, only one high speed camera was used in this study due to the limit of the present setup. However, it is still possible to reconstruct the bubble surface shape and identify the mode number from the observed oscillating cycles in the acquired images. Recently, Birkin et al.37 studied the non-inertial cavitation near an ultrasonic horn using a high speed camera. Their results provide a useful example of identifying the mode number from a fixed observing plane. Birkin et al. suggested that the observed non-spherical mode could be different from the actual one when the bubble oscillated near a surface. Therefore, extra care needs to be taken when reconstructing the surface mode from the imaging data alone. A similar methodology is employed in this paper to identify the mode numbers of the observed oscillating bubbles. The bubble, for example in Fig. 8(a), is seen to oscillate with the n ¼ 3 mode when it is translating toward the glass plate. Considering a 3D case, the bubble radius given in Eq. (1) needs to be expanded as Rðh; tÞ ¼ RðtÞ þ an;m ðtÞYn;m ðhÞ
Y3;0
¼ RðtÞ þ a3;0 ðtÞY3;0 ðhÞ; rffiffiffi 1 7 ð5 cos3 h 3 cos hÞ: ¼ 4 p
(11)
The natural frequency of a spherical mode is known as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fn ¼ ð1=2pÞ ðn 1Þðn þ 1Þðn þ 2Þr=q0 R30 ðn > 1Þ, and the mode frequency of f3 is 21.5 kHz. The duration of the six frames in Fig. 8(a) is 42 ls. This implies an oscillating frequency of 23.8 kHz, which is close to the predicted 21.5 kHz. Compared to the example given by Birkin et al. which focused on the bubble attached on the horn surface, the bubble shown in Fig. 8(a) is still separated from the glass
FIG. 9. Selected frames from a video of bubble oscillation near a surface at 17.2 kPa. The bubble initial radius is 54 lm. (a) The bubble oscillates with a non-spherical shape when it is approaching glass 1. (b) The arrival of the bubble on the surface. (c) The bubble oscillates with a non-spherical mode on the surface. (Scale bar: 200 lm, the period of one acoustic cycle: 21 ls.) Xi et al.: Study of non-spherical bubble oscillations
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surface at a certain distance. Therefore, it is likely that the mode number identification used here may not lead to a severer error for the bubbles away from the glass plate. It is worth mentioning that this agreement between the image data and the predicted mode number could only be applied to the bubbles oscillating with lower non-spherical modes, which can be relatively easily characterized from the images (for example, the triangle shape of n ¼ 3 mode). The mode identification given here is an approximation of the actual 3D one, and is used solely for the qualitatively interpretation of bubble shape changes near a surface. When the bubble arrives on the surface, the identification of mode number becomes more challenging. The bubble can keep its non-spherical oscillation as seen in Fig. 8(b). The duration of the six frames in Fig. 8(b) is about 42 ls which indicates the same mode frequency of 23.8 kHz as seen in Fig. 8(a). However, Figs. 8(b) and 8(c) show that the bubble shape is no longer a triangle one, instead, a nonregular shape. As seen in Secs. V B and V C, several nonspherical modes can be excited at the same time when the bubble is on the wall. A single dominating mode shown in the low pressure amplitude case [Figs. 7(a)–7(c)] is replaced by a combined effect of different oscillation modes. A further investigation on the complicated shape of a bubble on a wall is required in a future study, and hence, will not be discussed in detail in the paper. B. Bubble translation with a spherical shape
For spherical bubbles shown in Figs. 3–6, their pulsations synchronize with the driving frequency. The ratio of the maximum radius to the equilibrium radius increases with an increase of the external pressure amplitude. When the bubbles approach glass 1, the influence of the wall on the
bubble oscillation becomes more significant. Figure 10 shows the calculated amplitudes of non-spherical oscillation modes for three bubbles in Figs. 3–6. The distortion amplitudes of bubbles away from glass 1 [Figs. 10(a1), 10(b1), and 10(c1)] are simulated by Eqs. (2) and (8). Figures 10(a2), 10(b2), and 10(c2) display the distortion amplitudes of bubbles attached on the glass plate based on Eqs. (2) and (10). Following the approach of Versluis et al.,29 the initial perturbation an(0) is arbitrarily set to 1 nm, which is much smaller than the size of the bubbles used in the experiment and appears as a reasonable value of radial fluctuations. From Figs. 10(a1) and 10(a2), it now follows that the nonspherical oscillations of the 43 lm bubble can hardly be excited at 6.9 kPa, which holds for the situation before and after its arrival on glass 1. The amplitudes of all modes n ¼ 2 to 6 decay within a few ultrasonic driving cycles, and the higher modes disappear more rapidly (where the least stable mode is n ¼ 3). This outcome coincides with the observed stable spherical bubble oscillations as seen in Fig. 3. The neighborhood of a boundary is not only shifting a bubble’s “breathing mode” (n ¼ 0) resonance, but also the resonances (instability regions) of the higher surface modes n > ¼ 2 to lower frequencies or lower bubble radii, respectively. This can lead to the excitation of otherwise decaying non-spherical oscillations if the bubble approaches the boundary. In our case, this indeed occurs at the higher driving pressure amplitudes. At 12 kPa the n ¼ 3 mode starts to grow only when the bubble arrives on glass 1 [Fig. 10(b2)], which does not happen in the bulk liquid [Fig. 10(b1)]. The other mode amplitudes are still decaying at this driving pressure. Of course, the growth of an unstable mode does not saturate in the model, as it is based upon a linear ordinary differential equation. On the other hand, it is experimentally found that bubbles can oscillate at moderate pressure
FIG. 10. (Color online) Normalized bubble non-spherical pulsation amplitudes for a bubble of 43 lm radius. The mode number is selected from n ¼ 2–6. The non-spherical oscillation amplitudes are displayed for the bubble pulsating at a separation distance dwall ¼ 0.3 mm at 6.9 kPa (a1), 12 kPa (b1), and 17.2 kPa (c1). When the bubble is pulsating on glass 1, its non-spherical oscillation amplitudes are shown for 6.9 kPa (a2), 12 kPa (b2), and 17.2 kPa (c2).
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Xi et al.: Study of non-spherical bubble oscillations
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amplitudes non-spherically without breaking up. Such a phenomenon needs the inclusion of further effects such as nonlinear damping terms or mode-coupling to higher modes. However, this is beyond the scope of this consideration, and we are only concerned here with the small amplitude (linear) stability of the modes in the presence of a wall. Three possibilities that can be responsible for the observed non-spherical oscillation are recognized here. First, the bubble’s translation in the standing wave leads to a higher pressure amplitude at other positions in the acoustic field, in particular at the glass surface (see Fig. 1). The pressure amplitudes are calculated based on a one-dimensional transducer matrix model which were validated for the propose test cell in previous studies.34,35 In the case of Fig. 3, for example, the experimental pressure amplitude increases from 11.3 kPa at dwall ¼ 0.3 mm (the bubble-wall separation distance defined in Fig. 2) to 12 kPa on the glass surface. Second, the pressure threshold for exciting a bubble’s non-spherical oscillation mode is shifted to a lower level due to the wall boundary. This change of amplitude is exemplified in Fig. 11(a), which demonstrates the changes of pressure threshold to excite the modes n ¼ 2 to 5. The solid lines in Fig. 11(a) represent the pressure threshold of a free bubble (no resonance shift is considered for the bubble), while the dashed lines are the thresholds for a bubble attached on the wall (resonance frequency is shifted downward for the bubble). At the driving frequency, the pressure threshold for the mode n ¼ 3 decreases from 21 kPa (free bubble, solid line) to 5 kPa (bubble attached on a wall, dashed line), while the n ¼ 2 mode drops from 16 to 4 kPa. Figure 5 provides a good example of the influence of shifted resonance on the excitation of bubble non-spherical oscillation. Supposing the glass plate is removed from the system and the pressure profile does not change significantly, it can be seen from Fig. 11(a) that the bubble can maintain its spherical shape at 12 kPa which is lower than the threshold of n ¼ 2 and 3 modes (solid lines). On the other hand, in the presence of the wall, the bubble can experience non-spherical oscillation at the same location at the same amplitude due to the lower threshold of n ¼ 2 and 3 modes. This is in good agreement with the observed non-spherical oscillation amplitudes in Fig. 5. This effect is also demonstrated in Fig. 10(b2). It is worth mentioning that the nearby wall can also inhibit the growth of non-spherical oscillation at low amplitude. The bubble surface movement can be constrained when it oscillates on the wall as seen in Fig. 3. Third, the symmetric flow around a free bubble cannot be perfectly established any more for a bubble on the wall. Thus, the presence of glass 1 causes permanent nonspherical perturbations of the bubble wall which might establish growing surface modes. Another factor that can influence the bubble oscillation is the presence of neighboring bubbles. The coalescence of two bubbles creates a new large bubble with a larger radius R0. The relationship between bubble radii and pressure thresholds is shown in Fig. 11(b). The n ¼ 3 mode, for example, could be excited for the 43 lm bubble when it is alone, while a more complicated combination of modes n ¼ 4 to 6 (superposition of several modes) can be triggered when the J. Acoust. Soc. Am., Vol. 135, No. 4, April 2014
FIG. 11. (Color online) Pressure threshold for exciting non-spherical oscillation modes. (a) The thresholds for a bubble of radius of 43 lm as seen in Fig. 5. The pressure thresholds are calculated based on Eq. (3). Solid lines are pressure thresholds for a free bubble, while the dashed line is the pressure threshold for a bubble attached on the wall. The non-spherical modes are numbered from 2 to 5. (b) The thresholds for bubble radii ranging from 20 to 100 lm. The non-spherical modes are numbered from 2 to 5. The pressure thresholds are calculated based on Eq. (3). Solid lines represent pressure threshold of a free bubble, and the scattered lines are thresholds of a bubble attached to a wall.
bubble size grows to 60 lm after the coalescence. Moreover, a large perturbation on the bubble surface is seen during the bubble coalescence [Figs. 6(c) and 6(d)]. This could also contribute to the observed surface modes. C. Bubble translation with a non-spherical shape
In the sound field away from the glass surface, a bubble can oscillate with a non-spherical mode when its size or the external pressure amplitude exceeds a certain threshold. For a bubble of 54 lm radius (Fig. 7), the n ¼ 3 mode can be excited at 6.9 kPa as seen in Fig. 12(a1). The dominating n ¼ 3 mode can be maintained during the bubble’s translation. After arriving on the surface, the bubble experiences a transition period when its spherical oscillation is resumed from the n ¼ 3 mode. At the low pressure amplitude, the wall can temporarily inhibit the non-spherical oscillation. However, the presence of the wall limits the bubble surface movement. The spherical oscillation only Xi et al.: Study of non-spherical bubble oscillations
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FIG. 12. (Color online) Normalized bubble non-spherical pulsation amplitudes for a bubble of 54 lm radius. The mode number is selected from n ¼ 2 to 6. The non-spherical oscillation amplitudes are displayed for the bubble pulsating at 6.9 kPa (a1), 12 kPa (b1), and 17.2 kPa (c1). When the bubble is pulsating on glass 1, its non-spherical oscillation amplitudes are shown for 6.9 kPa (a2), 12 kPa (b2), and 17.2 kPa (c2).
lasts for a short period of time before the bubble starts the non-spherical oscillation again. It can be seen in Fig. 12(a2) that the n ¼ 3 mode experiences a growth as a function of time which is confirmed by the experimental observation. It is worth mentioning that the influence of the wall on the transition between spherical and non-spherical oscillations is only discernable at low pressure amplitude. For the bubble oscillating at a higher amplitude (Fig. 8), the wall can hardly restore the spherical oscillation when the bubble arrives on the wall surface. At a higher amplitude, the n ¼ 3 mode is still dominating when the bubble is away from the wall [Fig. 12(b1)]. Similar to the 6.9 kPa case, all spherical modes are triggered after the arrival of bubble on the surface. At 17.2 kPa, the n ¼ 3 mode is seen to be slightly larger than the other modes which, however, also increases with an increase of time instead of decreasing as seen in Figs. 12(a1) and 12(b1). The growth of all non-spherical modes are seen for all three amplitudes when the bubbles arrive on the surface. It is noticed from Figs. 12(a2), 12(b), and 12(c2) that the simulated mode numbers n ¼ 2 and 3 can be excited for the 54 lm radius bubble at the three amplitudes. With an increase of driving pressure amplitude, more higher modes can be triggered which complicated the bubble oscillation near the wall. VI. CONCLUSION
The excitation of non-spherical bubble dynamics near a surface in an acoustic standing wave field was investigated. High-speed recordings from an experiment were compared to numerical calculations based on the modified RayleighPlesset models. The excitation of non-spherical modes 1740
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requires the pressure amplitude to exceed a certain threshold for a given size of bubble. On the other hand, at a fixed pressure amplitude, different bubble sizes result in different oscillation modes. The neighborhood of a boundary is shifting a bubble’s linear (mode n ¼ 0) resonance, as well as the resonances (instability regions) of the higher surface modes n 2, to lower frequencies or lower bubble radii, respectively. Thus, for a bubble translating with a spherical shape, a nearby surface can lower the pressure threshold of non-spherical oscillation modes. Therefore, the bubble can shift from the spherical oscillation to a non-spherical one just by approaching a wall. The inverse effect is also possible, and the transition between non-spherical and spherical and non-spherical oscillations was observed in the experiment. An increase of pressure amplitude forced the experimental bubbles to undergo stronger surface shape deformations which were substantially enhanced by a neighboring surface. When a bubble approached the target surface, a shifted non-spherical oscillation pressure threshold can force the bubble to switch to another oscillation mode. The bubble oscillation mode is further complicated at a higher pressure amplitude, which can excite additional higher oscillation modes. All of the observed bubble oscillation mode transitions are following the same trend as the theoretical predictions. Understanding the bubble surface deformation near a wall is very important, because it influences the flow field around the bubble. The disturbed flow field is useful in many therapeutic and industrial applications using microbubbles, such as drug delivery, sonoporation, and ultrasonic cleaning, etc. The characteristics of the flow field will be investigated in the future. ACKNOWLEDGMENTS
X.X. and F.C. would like to acknowledge financial support from Lam Research AG (SEZ-Strasse 1, 9050 Villach, Xi et al.: Study of non-spherical bubble oscillations
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Austria). R.M. gratefully acknowledges financial support by the Austrian Federal Ministry of Economy, Family and Youth and the Austrian National Foundation for Research, Technology and Development. 1
D. Krefting, R. Mettin, and W. Lauterborn, “High-speed observation of acoustic cavitation erosion in multibubble systems,” Ultrason. Sonochem. 11, 119–123 (2004). 2 C. Ohl, M. Arora, R. Dijkink, V. Janve, and D. Lohse, “Surface cleaning from laser induced cavitation bubbles,” Appl. Phys. Lett. 89, 074102 (2006). 3 I. Akhatov, O. Lindau, A. Topolnikov, R. Mettin, N. Vakhitova, and W. Lauterborn, “Collapse and rebound of a laser-induced cavitation bubble,” Phys. Fluids 13, 2805–2819 (2001). 4 W. Lauterborn and H. Bolle, “Experimental investigations of cavitationbubble collapse in the neighborhood of a solid boundary,” J. Fluid Mech. 72, 391–399 (1975). 5 C. Ohl, T. Kurz, R. Geisler, O. Lindau, and W. Lauterborn, “Bubble dynamics, shock waves and sonoluminescence,” Philos. Trans. R. Soc., A 357, 269–294 (1999). 6 K. Suslick, “The chemical effects of ultrasound,” Sci. Am. 260, 80–86 (1989). 7 X. Xi, F. Yang, D. Chen, Y. Luo, D. Zhang, N. Gu, and J. Wu, “A targeting drug-delivery model via interactions among cells and liposomes under ultrasonic excitation,” Phys. Med. Biol. 53, 3251–3265 (2008). 8 C. Ohl, M. Arora, R. Ikink, N. De Jong, M. Versluis, M. Delius, and D. Lohse, “Sonoporation from jetting cavitation bubbles,” Biophys. J. 91, 4285–4295 (2006). 9 P. Prentice, A. Cuschieri, K. Dholakia, M. Prausnitz, and P. Campbell, “Membrane disruption by optically controlled microbubble cavitation,” Nat. Phys. 1, 107–110 (2005). 10 A. van Wamel, K. Kooiman, M. Harteveld, M. Emmer, F. ten Cate, M. Versluis, and N. de Jong, “Vibrating microbubbles poking individual cells: Drug transfer into cells via sonoporation,” J. Controlled Release 112, 149–155 (2006). 11 W. Kim, T. Kim, J. Choi, and H. Kim, “Mechanism of particle removal by megasonic waves,” Appl. Phys. Lett. 94, 081908 (2009). 12 A. Thiemann, T. Nowak, R. Mettin, F. Holsteyns, and A. Lippert, “Characterization of an acoustic cavitation bubble structure at 230 kHz,” Ultrason. Sonochem. 18, 595–600 (2011). 13 P. Marmottant and S. Hilgenfeldt, “Controlled vesicle deformation and lysis by single oscillating bubbles,” Nature 423, 153–156 (2003). 14 S. A. Suslov, A. Ooi, and R. Manasseh, “Nonlinear dynamic behavior of microscopic bubbles near a rigid wall,” Phys. Rev. E 85, 066309 (2012). 15 M. Strasberg, “The pulsation frequency of nonspherical gas bubbles in liquids,” J. Acoust. Soc. Am. 25, 536–537 (1953). 16 E. Payne, S. Illesinghe, A. Ooi, and R. Manasseh, “Symmetric mode resonance of bubbles attached to a rigid boundary,” J. Acoust. Soc. Am. 118, 2841–2849 (2005). 17 A. Doinikov, L. Aired, and A. Bouakaz, “Dynamics of a contrast agent microbubble attached to an elastic wall,” IEEE Trans. Med. Imaging 31, 654–662 (2012). 18 A. Doinikov, L. Aired, and A. Bouakaz, “Acoustic scattering from a contrast agent microbubble near an elastic wall of finite thickness,” Phys. Med. Biol. 56, 6951–6967 (2011). 19 A. Doinikov, L. Aired, and A. Bouakaz, “Acoustic response from a bubble pulsating near a fluid layer of finite density and thickness,” J. Acoust. Soc. Am. 129, 616–621 (2011).
J. Acoust. Soc. Am., Vol. 135, No. 4, April 2014
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V. Garbin, D. Cojoc, E. Ferrari, E. Di Fabrizio, M. Overvelde, S. van Der Meer, N. de Jong, D. Lohse, and M. Versluis, “Changes in microbubble dynamics near a boundary revealed by combined optical micromanipulation and high-speed imaging,” Appl. Phys. Lett. 90, 114103 (2007). 21 M. Plesset, “On the stability of fluid flows with spherical symmetry,” J. Appl. Phys. 25, 96–98 (1954). 22 A. Prosperetti and Y. Hao, “Modelling of spherical gas bubble oscillations and sonoluminescence,” Philos. Trans. R. Soc., A 357, 203–223 (1999). 23 S. Hilgenfeldt, D. Lohse, and M. Brenner, “Phase diagrams for sonoluminescing bubbles,” Phys. Fluids 8, 2808–2826 (1996). 24 J. Loughran, R. Eckersley, and M. Tang, “Modeling non-spherical oscillations and stability of acoustically driven shelled microbubbles,” J. Acoust. Soc. Am. 131, 4349–4357 (2012). 25 A. Maksimov and T. Leighton, “Pattern formation on the surface of a bubble driven by an acoustic field,” Philos. Trans. R. Soc., A 468, 57–75 (2012). 26 S. Zhao, K. Ferrara, and P. Dayton, “Asymmetric oscillation of adherent targeted ultrasound contrast agents,” Appl. Phys. Lett. 87, 134103 (2005). 27 C. Caskey, D. Kruse, P. Dayton, T. Kitano, and K. Ferrara, “Microbubble oscillation in tubes with diameters of 12, 25, and 195 microns,” Appl. Phys. Lett. 88, 033902 (2006). 28 H. Vos, B. Dollet, M. Versluis, and N. de Jong, “Nonspherical shape oscillations of coated microbubbles in contact with a wall,” Ultrasound Med. Biol. 37, 935–948 (2011). 29 M. Versluis, D. Goertz, P. Palanchon, I. Heitman, S. van der Meer, B. Dollet, N. de Jong, and D. Lohse, “Microbubble shape oscillations excited through ultrasonic parametric driving,” Phys. Rev. E 82, 026321 (2010). 30 M. Longuet-Higgins, “Monopole emission of sound by asymmetric bubble oscillations. Part 1. Normal modes,” J. Fluid Mech. 201, 525–541 (1989). 31 A. Doinikov, “Translational motion of a bubble undergoing shape oscillations,” J. Fluid Mech. 501, 1–24 (2004). 32 P. Wilcox, R. Monkhouse, P. Cawley, M. Lowe, and B. Auld, “Development of a computer model for an ultrasonic polymer film transducer system,” NDT&E Int. 31, 51–64 (1998). 33 M. Hill, Y. Shen, and J. Hawkes, “Modelling of layered resonators for ultrasonic separation,” Ultrasonics 40, 385–392 (2002). 34 X. Xi, F. Cegla, M. Lowe, A. Thiemann, T. Nowak, R. Mettin, F. Holsteyns, and A. Lippert, “Study on the bubble transport mechanism in an acoustic standing wave field,” Ultrasonics 51, 1014–1025 (2011). 35 X. Xi, F. Cegla, R. Mettin, F. Holsteyns, and A. Lippert, “Collective bubble dynamics near a surface in a weak acoustic standing wave field,” J. Acoust. Soc. Am. 132, 37–47 (2012). 36 A. Francescutto and R. Nabergoj, “Pulsation amplitude threshold for surface waves on oscillating bubbles,” Acustica 41, 215–220 (1978). 37 P. Birkin, D. Offin, C. Vian, T. Leighton, and A. Maksimov, “Investigation of noninertial cavitation produced by an ultrasonic horn,” J. Acoust. Soc. Am. 130, 3297–3308 (2011). 38 M. A. Ainslie and T. G. Leighton, “Review of scattering and extinction cross-sections, damping factors, and resonance frequencies of a spherical gas bubble,” J. Acoust. Soc. Am. 130, 3184–3208 (2011). 39 Y. A. Kobelev and L. A. Ostrovskii, “Acousto-electrostatic analogy and gas bubbles interaction in liquid,” Sov. Phys. Acoust. 30, 715–716 (1984). 40 H. Oguz and A. Prosperetti, “Bubble oscillations in the vicinity of a nearly plane free surface,” J. Acoust. Soc. Am. 87, 2085–2092 (1990). 41 C. Pozrikidis, Boundary Integral and Singularity Methods for Linearized Viscous Flow (Cambridge University Press, Cambridge, 1992), Chap. 7. 42 A. Doinikov, “Translational motion of a spherical bubble in an acoustic standing wave of high intensity,” Phys. Fluids 14, 1420–1425 (2002). 43 R. Manasseh and A. Ooi, “Frequencies of acoustically interacting bubbles,” Bubble Sci., Eng., Technol. 1, 58–74 (2009).
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Robert Mettin Christian Doppler Laboratory for Cavitation and Micro-Erosion, Drittes Physikalisches Institut, Georg-August-Universit€ at G€ ottingen, Friedrich-Hund-Platz 1, 37077 G€ ottingen, Germany
Frank Holsteyns and Alexander Lippert Lam Research AG, SEZ-Strasse 1, 9050 Villach, Austria
(Received 10 April 2013; revised 13 January 2014; accepted 22 January 2014) The interaction of acoustically driven bubbles with a wall is important in many applications of ultrasound and cavitation, as the close boundary can severely alter the bubble dynamics. In this paper, the non-spherical surface oscillations of bubbles near a surface in a weak acoustic standing wave field are investigated experimentally and numerically. The translation, the volume, and surface mode oscillations of bubbles near a flat glass surface were observed by a high speed camera in a standing wave cell at 46.8 kHz. The model approach is based on a modified Keller-Miksis equation coupled to surface mode amplitude equations in the first order, and to the translation equations. Modifications are introduced due to the adjacent wall. It was found that a bubble’s oscillation mode can change in the presence of the wall, as compared to the bubble in the bulk liquid. In particular, the wall shifts the instability pressure thresholds to smaller driving frequencies for fixed bubble equilibrium radii, or to smaller equilibrium radii for fixed excitation frequency. This can destabilize otherwise spherical bubbles, or stabilize bubbles undergoing surface oscillations in the bulk. The bubble dynamics observed in experiment demonstrated the same trend as the theoretical results. C 2014 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4864461] V PACS number(s): 43.25.Yw, 43.20.Ks, 43.35.Yb, 43.25.Qp [TGL]
I. INTRODUCTION
Understanding the interaction of acoustically driven bubbles and nearby solid or flexible boundaries is essential in many applications of ultrasound and acoustic cavitation. The most prominent effects of strong bubble oscillations at solids are material erosion1 and cleaning of attached contaminations.2 Both are typically associated with high pressure peaks and strong shear flows occurring during bubble collapse3 and possible liquid jet impact.4,5 Other observed physical and chemical effects, like surface activation or enhanced chemical reactions of solids in liquids,6 are due to extremes in pressure or temperature caused by close or attached bubbles during collapse. The strong shear flow associated with heavy collapse and jetting also causes substantial forces on flexible boundaries, where tissue and cell membranes are of particular interest.7,8 A typical and standard application is the disintegration of cells by ultrasonic cavitation.9 On the other hand, the flow field of only weakly or moderately oscillating acoustically driven bubbles has proven to be effective in more delicate applications like sonoporation10 or removal of small particles from structured substrates.11,12 Here, pressure shocks and fast jet impact can be counterproductive or harmful, and should be avoided. Nevertheless, significant perturbations of substrate or membrane boundary a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]
J. Acoust. Soc. Am. 135 (4), April 2014
Pages: 1731–1741
layers seem achievable, and by controlling the bubble oscillations and the mediated shear at the boundaries, driven bubbles might be engineered as a tool.13 Thus, it is necessary to consider in detail the influence of the boundary on weak to moderate bubble oscillations and the associated flow field surrounding the bubbles. While existing literature is related to this topic, there are still many open questions. In this work, we address the question of the onset of non-spherical mode oscillations of driven single bubbles directed toward and reaching a solid surface which is investigated experimentally and compared to theory. Assuming a bubble remains spherical to first order, a neighboring wall could force the bubble to change its behavior when it is oscillating on the surface. Suslov et al.14 calculated the nonlinear response of micro-bubbles near a rigid wall based on a modified Keller-Miksis equation. The resonance frequency of a bubble that oscillates on a wall was simulated by Strasberg15 and Payne et al.16 It has been recognized that the resonance frequency of a bubble near a rigid wall is lower than that of a free bubble. However, Doinikov et al. found out that the elasticity of a wall cannot be neglected. They proposed several modified Rayleigh-Plesset equations to explain the bubble dynamics on elastic walls of different properties.17–19 Experimental work was also carried out in the past to understand the spherical bubble behavior near a wall. Garbin et al.20 observed the change of bubble dynamics near a boundary using a high speed camera system.
0001-4966/2014/135(4)/1731/11/$30.00
C 2014 Acoustical Society of America V
1731
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Besides the spherical oscillation, a bubble can also oscillate non-spherically in an ultrasound field. In a bulk medium, the spherical shape stability was first modeled by Plesset.21 Later on, the viscous effect in the vicinity of a bubble surface was considered by Prosperetti and Hao.22 By only taking the local vorticity into account, Hilgenfeldt et al.23 studied the shape instabilities and diffusive instabilities for a single sonoluminescence bubble. Loughran et al.24 considered the shell properties of Ultrasound Contrast Agents (UCAs), and calculated the non-spherical behavior of shelled micro-bubbles. Maksimov and Leighton25 developed a model that explains the selection of the regular pattern on the bubble wall. In contrast, the influence of a wall on the nonspherical bubble oscillation is still not completely clear. Also, due to the complicated bubble-wall interaction, little experimental work on non-spherical bubble oscillation can be found in the literature. Recently, the nonspherical behaviors of UCAs within micro-vessels were reported by Zhao et al.26 and Caskey et al.,27 respectively. Vos et al.28 measured the non-spherical shapes of UCAs near cellulose walls, and analyzed the corresponding shear stress exerted on the boundary. Versluis et al.29 observed the shape oscillations of micro-bubbles excited through parametric driving without considering the wall effects. In these previous experimental studies, non-spherical bubbles were normally placed at the monitoring positions prior to the observations. In many real life applications, bubbles need to be transported to a target area because they are generated far from the surface. It is well-known that a bubble’s non-spherical oscillation modes interact with each other during its translation. In a bulk medium, this coupling effect between different modes was theoretically investigated by Longuet-Higgins30 and Doinikov.31 Nevertheless, little is understood about the translation of a non-spherical bubble near a wall. The change of nonspherical oscillation modes before and after the bubble’s arrival on a boundary has not been investigated. This is due to the lack of a reliable experimental configuration that can control the bubble oscillation modes in the vicinity of a boundary. In this paper, we report the influence of the translation toward a neighboring wall on the nonspherical bubble oscillations. In the experiments, the bubble motion control was achieved using a multi-layered stack that was designed based on a one-dimensional matrix transducer model.32,33 The test rig has been verified as an effective tool to control single and multi-bubble transportation in an acoustic standing wave field.34,35 Numerically, the dynamics of non-spherical bubbles near a wall are analyzed based on modified Keller-Miksis equations, a dynamical equation, and a pair of translation equations. This paper is organized as follows: Section II shows the experimental setup. In Sec. III, the theoretical background of non-spherical bubble dynamics near a surface is presented. The observed bubble oscillation modes are demonstrated in Sec. IV, and are discussed in Sec. V. Conclusions are drawn in Sec. VI. 1732
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II. EXPERIMENTAL CONFIGURATION
A schematic diagram of the experimental setup is shown in Fig. 1. Bubbles of radii ranging from 10 to 80 lm were generated by an electrolysis method and contained hydrogen. The bubble motion was manipulated within a multi-layered stack that consists of an ultrasonic transducer and a water chamber with side windows. The acoustic field in the water corresponds well to a one-dimensional standing wave in the x direction (Fig. 1). The transducer was operated at a driving frequency of 46.8 kHz. At this frequency, the sizes of the test bubbles are much smaller than the acoustic wavelength. A high speed camera system (FastCam SA5, Photron, San Diego, CA) was used to visualize the bubble dynamics near a surface at various frame rates up to 525 000 frames/s. A flat target glass plate of 0.1 mm thickness (VWR, United Kingdom) was inserted parallel to the wave front at x ¼ 4 mm (the origin of the coordinate system is selected at the transducer-water boundary, x ¼ 0, y ¼ 0, z ¼ 0). The water chamber was terminated by a similar glass plate against air at x ¼ 8 mm. More details of the test rig were reported in previous work.34,35
III. THEORY
When the acoustic pressure amplitude reaches a certain value, a driven bubble can start to oscillate non-spherically in the sound field. It is well-known that the non-spherical modes are related to their spherical counterpart. The spherical oscillation can be modeled by different radial oscillation equations depending on the bubble-wall separation distance. In this section, two cases of non-spherical oscillations are considered: A bubble translating toward a wall, and a bubble oscillating on a boundary. To calculate the non-spherical modes, let us first introduce an initial small perturbation on a bubble’s surface, which is given by
FIG. 1. (Color online) A schematic diagram of the test rig for studying the bubble behavior. The high speed camera system is used to focus on the field of view at 525 000 frames/s. The bubble is transported from the injection point (x ¼ 5 mm) to glass 1 (x ¼ 4 mm) within an acoustic standing wave field. The predicted pressure profile in the water layer is shown. Xi et al.: Study of non-spherical bubble oscillations
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Rðh; tÞ ¼ RðtÞ þ an ðtÞYn ðhÞ;
(1) x0free
where R(t) is the time-varying bubble radius, Yn(h) is the spherical harmonic of the nth order with amplitude of an(t). Including the local viscous effect, the distortion amplitude an(t) takes the form of23 a€n ðtÞ þ Bn ðtÞa_ n ðtÞ An ðtÞan ðtÞ ¼ 0;
d ðn 1Þðn þ 2Þ þ 2nðn þ 2Þðn 1Þ ; R _ 3R 2 d þ 2 ðn þ 2Þð2n þ 1Þ 2nðn þ 2Þ2 ; Bn ðtÞ ¼ R R R
(2)
where r is the surface tension, q is the liquid density, is the kinematic viscosity, x is the angular frequency, and bn ¼ (n 1)(n þ 1)(n þ 2). The mode coupling between the mode n ¼ 0 (radial) to higher modes n ¼ 2,3,…, etc., and the mode n ¼ 1 to n ¼ 0 is considered. The coupling between higher modes n ¼ 2,3,…, etc., is neglected. To excite a non-spherical oscillation, the external pressure needs to exceed a certain threshold Pthreshold,36,37
Pthreshold
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u ðs 1Þ2 þ 4p h ¼ qR20 u ; 2 u t 3 1 þ q2 s þ 2p þ 2 l þ 2 2
h ¼ ðx20 x2 Þ2 þ ð2btot xÞ2 ; 4ðn 1Þðn þ 1Þ þ ðn þ 2Þr ; qx2 R30 " #2 2ðn þ 2Þð2n þ 1Þ p¼ ; xR20 s¼
q¼
6ðn þ 2Þ ; xR20
(4)
where c is the polytropic exponent of the gas inside the bubble, and Pv is the vapor pressure in the liquid. Previous studies showed that the resonance frequency x0free is shifted downward when the bubble oscillates on a rigid wall. The lowered resonance frequency is given by16 (5)
where dwall is the separation distance between a bubble and the wall. It is worth mentioning that Payne et al.16 only considered the monopole component in analyzing the interaction between the original bubble and its mirror counterpart. However, when the separation distance is comparable to the bubble size, the multipole expansion terms need to be taken into account for a more accurate solution. Moreover, when the bubble size is smaller than the wavelength, the pressure inside the bubble is constant because the equilibrium pressure P0 is larger than that from the surface tension. The bubble surface can be treated as equipotential. This analogy with electrostatics was discussed by Kobelev and Ostrovskii.39 A similar method was also used by Oguz and Prosperetti40 for studying the bubble dynamics near a free surface. Since the theoretical predictions are only used here to describe the qualitative trend of bubble behavior and the use of Payne’s result will not cause severe errors as seen in the following experimental results, the investigation of an accurate analytical solution of the flow surround the bubble will not be discussed. For a bubble far away from the wall (dwall R0), the influence of the wall is negligible (x0 ¼ x0free). On the other hand, for a bubble that is attached on a rigid wall, Eq. (5) indicates that x0 is lower pffiffiffiffiffiffiffithan the free bubble resonance frequency x0 ¼ x0free = 1:5 which is not very different from the solutionpffiffiffiffiffiffiffiffiffi obtained by Kobelev and Ostrovskii39 x0 ¼ x0free log2 . The damping effect of a bubble has been extensively studied in the literaturep(see Ref. 38). A typical length scale ffiffiffiffiffiffiffiffiffiffiffi for viscous effect d ¼ 2=x is introduced here. When the length scale is relatively thin, the damping factor for the non-spherical modes is25
(3)
where R0 is the equilibrium bubble radius which is much smaller than the acoustic wavelength (R0 k). btot is the total damping factor and x0 is the angular resonance frequency. It is worth mentioning that Eq. (3) is related to two resonance factors (x0 and btot). The presence of a nearby wall could introduce correction terms into these two factors. Therefore, even small changes in x0 and btot can result in a significant difference in the prediction of pressure threshold. The angular frequency x0 has been thoroughly examined in the literature (see Ref. 38, for example). For an un-coated bubble, x0free is given by J. Acoust. Soc. Am., Vol. 135, No. 4, April 2014
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2r 2r 3c P0 þ Pv þ Pv ; R0 R0
x0free x0 ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; R0 1þ 2dwall
R€ b r 2 R_ An ðtÞ ¼ ðn 1Þ n 3 3 R R qR
rffiffiffiffi ! R d ¼ min ; x 2n
1 ¼ pffiffiffi R0 q
bn ¼ ðn þ 2Þð2n þ 1Þ
: R20
(6)
The total damping factor is a combined effect of viscous, thermal, and acoustic damping. The damping factor for the breathing mode n ¼ 0 is estimated by a linear analysis as25 rffiffiffiffiffiffi x2 R0 2 x0wall D þ 2 þ 3ðc 1Þ b0 ; (7) 2c 2R0 2x R0 where D is the diffusivity coefficient. When a bubble oscillates near a wall, the boundary can be replaced by a mirror bubble on the other side of the wall. Xi et al.: Study of non-spherical bubble oscillations
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The mirror bubble oscillates in phase with the original one at the same amplitude in the presence of a rigid wall.41 As seen in Eq. (2), the amplitude of the original bubble’s nonspherical oscillation is closely related to its spherical radius R(t). R(t) can be obtained from the two radial oscillation equations depending on the bubble-wall separation distance (dwall). Figures 2(a) and 2(b) show the cases when a bubble translates toward an elastic boundary and when it oscillates on the surface, respectively. In both cases, dwall is assumed to be larger than R0. The bubble radial oscillation equation is welldocumented in the literature. Recently, Doinikov et al.17,19 pointed out that the oscillation of a bubble near an elastic wall depends on the separation distance (dwall), wall density (qwall), and thickness (Lwall). Donikov et al. neglected the shear strain of the wall because the motion of a soft wall is mainly caused by the compressive liquid movement. In the present case, however, such a simplification may not be valid because the shear wave velocity of glass is higher than soft tissues. Hence, the normal flow velocity on the glass surface is almost zero which means the glass plate can be treated as a rigid wall. The radial oscillation of a bubble near a rigid wall is governed by a modified Keller-Miksis equation.14,34,35 The radial equation proposed by Xi et al.35 considers the collective behavior of a group of bubbles. In the model, the bubble-wall interaction is represented by a bubble-imaginary-bubble system. The modified Keller-Miksis equation is hence given by 3 R_ _ 2 R_ € 1 RR þ R 2 2c c x_ 2 1 R 1 R_ 2 € 1þ ¼ ð2R_ R þ R2 RÞ; Psc þ P_ sc þ q qc 4 2dwall c c 2r R30 a3 2r 4gR_ P0 Pex ; Psc ¼ P0 þ 3 3 R a R0 R R Pex ¼ Pa sinðxtÞsinðkdnode Þ;
(8)
where c is the sound velocity in the liquid, and Psc is the scattered pressure. The bubble center position is given by (x, y) in a two-dimensional system. P0 is the hydrostatic pressure, a is the van der Waals hard core, and Pex is the external driving pressure which takes the form of a standing wave. Pa is the pressure amplitude at the antinode, k is the wave number, and dnode is the separation distance between a bubble center and a pressure node. The second term on the right-hand side of Eq. (8) ðx_ 2 =4Þ represents the coupling between the translation and radial oscillation, while the last term is the contribution of the imaginary bubble on the radial oscillation of the original one. The translation equations governing the bubble motion in the x and y axes are34,35,42,43 3R_ x_ 3Fx ¼ ; 2pqR3 R 3Fy 3R_ y_ ¼ ; y€ þ 2pqR3 R x€ þ
(9)
where Fx and Fy are external forces in the x and y axes, respectively. Figure 2(b) shows another case when a bubble oscillates on a boundary. In this case, the bubble surface displacement is substantially influenced by the wall. By removing the translation term in the modified Keller-Miksis equation, the radial equation for a bubble on the glass surface is 3 R_ _ 2 R_ € RR þ 1 R 2 2c c 1 1 R_ R d 2 € 1þ þ ¼ ð2R_ R þ R2 RÞ: Psc q 2dwall c c dt (10) The amplitude of non-spherical modes an can be solved by combining Eqs. (2), (8), and (10) together. It needs to be pointed out that non-spherical oscillation is a complicated
FIG. 2. The spherical oscillation of a bubble (a) at a separation distance dwall from an elastic boundary, and (b) on the boundary. A mirror bubble is introduced to replace the boundary. This mirror bubble oscillates in phase with the original bubble near a rigid wall.
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Xi et al.: Study of non-spherical bubble oscillations
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process, and the solved an is only an approximation of the actual non-spherical behavior. Nevertheless, the aim of this study is to qualitatively understand the changes of bubble non-spherical modes near a wall. As seen later in this paper, the above method provides a qualitative insight into the bubble non-spherical modes. The development of an improved non-spherical model is planned in the future. IV. RESULTS
The values of the parameters used in this study are f ¼ 46.8 kHz, q ¼ 1000 kg/m3, qwall ¼ 2230 kg/m3, P0 ¼ 101.3 kPa, c ¼ 1480 m/s, r ¼ 0.072 N/m, c ¼ 1, g ¼ 0.001 Pa*s. The dimensions of objects in the videos were calibrated with a standard 300 lm wide stick. The pressure amplitude was calculated based on a onedimensional matrix transducer model, and was verified experimentally.32,33 As seen in Fig. 1, the sound pressure amplitude monotonically decreases from x ¼ 0 mm at the water-transducer interface to x ¼ 8 mm at the water-glass contact surface. In Secs. IV A and IV B, the pressure amplitudes that are discussed refer to the pressure on the righthand side surface of glass 1 (x ¼ 4.1 mm). A. Spherical bubble oscillation near a surface
At 6.9 kPa, a bubble of 43 lm mean radius translates toward glass 1 as shown in Fig. 3(a). The bubble maintains a spherical shape at this pressure amplitude and its radius periodically varies between 41 and 45 lm. The separation distance between the bubble and the boundary is 0.3 mm. Since the driving frequency is set as 46.8 kHz, the time of one acoustic cycle is 21 ls. The three frames in Fig. 3(a) represent the bubble oscillation over about one acoustic cycle. After 11 ms, the bubble arrives onto the surface and still keeps its spherical shape [Fig. 3(b)]. The maximum radius, however, increases from 45 to 50 lm, while the minimum radius is lowered from 41 to 39 lm. It is worth mentioning here that a nearby wall can distort a bubble’s surface when the bubble arrives on the wall. The spherically symmetric flow field around a free bubble cannot be established due to the boundary. At pressures below the threshold for parametric excitation of surface waves, the bubble volumetric oscillation is not spherically symmetric. However, the asymmetric shape is not visible from the
FIG. 3. Selected frames from a video of bubble oscillation near a surface at 6.9 kPa. The bubble mean radius is 43 lm. (a) The bubble oscillates with a spherical mode when approaching glass 1. The three frames represent the bubble oscillation over one acoustic cycle. (b) The bubble keeps its spherical shape when sitting on the glass surface. The bubble is supposed to touch glass 1 at a contact point. The scale bar represents 200 lm. The time of one acoustic cycle is 21 ls. J. Acoust. Soc. Am., Vol. 135, No. 4, April 2014
experimental results because it is small (proportional to the amplitude of the radial oscillations). When the pressure is above the instability threshold, surface waves are stronger than the volumetric oscillation result, and the surface waves are clearly seen in the photographs. At the same pressure amplitude, for a similar sized bubble, its spherical oscillation can be switched to a nonspherical one after merging with a neighboring bubble. In Fig. 4(a), a bubble (radius of 41 lm) oscillates spherically during its translation to glass 1. The spherical oscillation is disturbed by the arrival of another nearby bubble (radius of 28 lm) at 88 ms. The coalescence of the two bubbles is clearly seen in Fig. 4(b). The newly formed bubble (radius of 60 lm) then starts to oscillate with a non-spherical mode on the surface as shown in Fig. 4(c). At a higher pressure amplitude, 12 kPa, a bubble of 43 lm radius oscillates with a spherical shape where the bubble radius varies between 38 and 48 lm [Fig. 5(a)]. When the bubble arrives onto the surface in Fig. 5(b), it starts to lose its symmetric shape, and gradually shifts to a nonspherical oscillation mode. The non-spherical shape is finally formed in Fig. 5(c). The bubble starts to stretch in the horizontal direction where it reaches a maximum extension and then shrinks. The compression of the surface shape in the horizontal direction results in a growth of bubble in the vertical direction. When the bubble surface grows to the maximum in the vertical direction, its shape falls back to the original state and then repeats the whole process again. The influence of the wall on the oscillation of a bubble translating with a spherical shape is more substantial at 17.2 kPa. Initially, a spherical bubble translates in the sound field as seen in Fig. 6(a). After 1.6 ms, the bubble approaches glass 1 at a faster speed. Immediately, the bubble starts to change its shape from spherical to elliptical even without touching the glass surface. The shape change becomes more dramatic after the arrival of the bubble on the glass surface. The elongation of the bubble shape in the vertical direction eventually leads to a split of the bubble into two parts which
FIG. 4. Selected frames from a video of bubble oscillation near a surface at 6.9 kPa. The initial bubble radius is 41 lm. (a) The bubble oscillates without non-spherical modes when it is approaching glass 1. (b) A second bubble approaches the target and merges with the first one. (c) The newly formed bubble of 60 lm radius oscillates with a non-spherical shape. (Scale bar: 200 lm, the period of one acoustic cycle: 21 ls.) Xi et al.: Study of non-spherical bubble oscillations
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FIG. 5. Selected frames from a video of bubble oscillation near a surface at 12 kPa. The initial bubble radius is 43 lm. (a) The bubble oscillates spherically when it is approaching glass 1. (b) The bubble arrives on the surface (c) The bubble oscillates with a non-spherical shape of mode ¼ 3. (Scale bar: 200 lm, the period of one acoustic cycle: 21 ls.)
then merge back together to form a new bubble. Two subsequent coalescences of the target bubble with two other neighboring bubbles are also shown in Figs. 6(c) and 6(d), respectively. The newly formed bubble oscillates with a much more complicated mode and completely loses its spherical shape. B. Non-spherical bubble oscillation near a surface
On the other hand, a bubble can translate with a nonspherical shape when its size exceeds a certain threshold. At 6.9 kPa, for example, a bubble of 54 lm radius oscillates non-spherically when it moves toward glass 1 [Fig. 7(a)]. The triangle shape indicates that the bubble pulsates at the third mode during its translation. This triangle shape shifts back to a spherical one after the bubble’s arrival on the glass surface in Fig. 7(b). The temporary spherical shape, however, starts to grow back to a non-spherical one after a few acoustic cycles. The asymmetric surface shape is finally formed in Fig. 7(c). At 12 kPa, a bubble of 52 lm radius also shows a nonspherical oscillation behavior when it travels toward glass 1.
FIG. 6. Selected frames from a video of bubble oscillation near a surface at 17.2 kPa. The initial bubble radius is 43 lm. (a) The bubble oscillates with a spherical shape when it is approaching glass 1. (b) The bubble arrives on the surface. (c) Coalescence of the bubble with a nearby bubble. (d) A second coalescence of two bubbles on the surface. (Scale bar: 200 lm, the period of one acoustic cycle: 21 ls.) 1736
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FIG. 7. Selected frames from a video of bubble oscillation near a surface at 6.9 kPa. The initial bubble radius is 54 lm. (a) The bubble oscillates with a non-spherical shape (n ¼ 3) when it is approaching glass 1. (b) The bubble shape returns to a spherical one after its arrival on the glass surface. (c) The bubble oscillates with a non-spherical shape again. (Scale bar: 200 lm, the period of one acoustic cycle: 21 ls.) Xi et al.: Study of non-spherical bubble oscillations
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A similar triangle shape is seen in Fig. 8(a). Unlike the low pressure case, the bubble surface immediately changes to a more complicated shape after its arrival on the glass surface. This non-spherical shape keeps evolving throughout the following time as shown in Figs. 8(b) and 8(c). The identification of the non-spherical mode, however, is difficult to be given here owing to the limited resolution of the frames. Similarly, a bubble of 56 lm radius experiences a strong oscillation at 17.2 kPa in Fig. 9. An irregular surface shape is seen for this bubble during its translation. Glass 1 forces the bubble to change its shape even before it is hitting on the surface. The non-spherical shape becomes more substantial when the bubble starts to oscillate on the surface [Figs. 9(b) and 9(c)]. V. DISCUSSION OF THE EXPERIMENTAL RESULTS
As seen in Sec. IV, a bubble can experience various oscillation modes under different conditions. In this section, the observed non-spherical bubble oscillation mode is explained by the model shown in Sec. III. The influence of external pressure amplitude, bubble size, boundary condition, and neighboring bubbles on the bubble oscillation mode is discussed. A. Mode number identification
The images shown in Sec. IV demonstrate the various bubble oscillation modes near a wall under different external conditions. A bubble oscillating with higher modes is a complicated three-dimensional (3D) object. It is possible that the higher surface modes can project different images on different observing planes. Ideally, the reconstruction of the bubble surface modes requires the employment of several high
FIG. 8. Selected frames from a video of bubble oscillation near a surface at 12 kPa. The bubble initial radius is 54 lm. (a) The bubble oscillates with a non-spherical shape when it is approaching glass 1. (b) The arrival of the bubble on the surface. (c) The bubble oscillates with a non-spherical shape on the surface. (Scale bar: 200 lm, the period of one acoustic cycle: 21 ls.) J. Acoust. Soc. Am., Vol. 135, No. 4, April 2014
speed cameras that are located at different locations in order to capture the surface changes from all possible angles. Admittedly, only one high speed camera was used in this study due to the limit of the present setup. However, it is still possible to reconstruct the bubble surface shape and identify the mode number from the observed oscillating cycles in the acquired images. Recently, Birkin et al.37 studied the non-inertial cavitation near an ultrasonic horn using a high speed camera. Their results provide a useful example of identifying the mode number from a fixed observing plane. Birkin et al. suggested that the observed non-spherical mode could be different from the actual one when the bubble oscillated near a surface. Therefore, extra care needs to be taken when reconstructing the surface mode from the imaging data alone. A similar methodology is employed in this paper to identify the mode numbers of the observed oscillating bubbles. The bubble, for example in Fig. 8(a), is seen to oscillate with the n ¼ 3 mode when it is translating toward the glass plate. Considering a 3D case, the bubble radius given in Eq. (1) needs to be expanded as Rðh; tÞ ¼ RðtÞ þ an;m ðtÞYn;m ðhÞ
Y3;0
¼ RðtÞ þ a3;0 ðtÞY3;0 ðhÞ; rffiffiffi 1 7 ð5 cos3 h 3 cos hÞ: ¼ 4 p
(11)
The natural frequency of a spherical mode is known as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fn ¼ ð1=2pÞ ðn 1Þðn þ 1Þðn þ 2Þr=q0 R30 ðn > 1Þ, and the mode frequency of f3 is 21.5 kHz. The duration of the six frames in Fig. 8(a) is 42 ls. This implies an oscillating frequency of 23.8 kHz, which is close to the predicted 21.5 kHz. Compared to the example given by Birkin et al. which focused on the bubble attached on the horn surface, the bubble shown in Fig. 8(a) is still separated from the glass
FIG. 9. Selected frames from a video of bubble oscillation near a surface at 17.2 kPa. The bubble initial radius is 54 lm. (a) The bubble oscillates with a non-spherical shape when it is approaching glass 1. (b) The arrival of the bubble on the surface. (c) The bubble oscillates with a non-spherical mode on the surface. (Scale bar: 200 lm, the period of one acoustic cycle: 21 ls.) Xi et al.: Study of non-spherical bubble oscillations
1737
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surface at a certain distance. Therefore, it is likely that the mode number identification used here may not lead to a severer error for the bubbles away from the glass plate. It is worth mentioning that this agreement between the image data and the predicted mode number could only be applied to the bubbles oscillating with lower non-spherical modes, which can be relatively easily characterized from the images (for example, the triangle shape of n ¼ 3 mode). The mode identification given here is an approximation of the actual 3D one, and is used solely for the qualitatively interpretation of bubble shape changes near a surface. When the bubble arrives on the surface, the identification of mode number becomes more challenging. The bubble can keep its non-spherical oscillation as seen in Fig. 8(b). The duration of the six frames in Fig. 8(b) is about 42 ls which indicates the same mode frequency of 23.8 kHz as seen in Fig. 8(a). However, Figs. 8(b) and 8(c) show that the bubble shape is no longer a triangle one, instead, a nonregular shape. As seen in Secs. V B and V C, several nonspherical modes can be excited at the same time when the bubble is on the wall. A single dominating mode shown in the low pressure amplitude case [Figs. 7(a)–7(c)] is replaced by a combined effect of different oscillation modes. A further investigation on the complicated shape of a bubble on a wall is required in a future study, and hence, will not be discussed in detail in the paper. B. Bubble translation with a spherical shape
For spherical bubbles shown in Figs. 3–6, their pulsations synchronize with the driving frequency. The ratio of the maximum radius to the equilibrium radius increases with an increase of the external pressure amplitude. When the bubbles approach glass 1, the influence of the wall on the
bubble oscillation becomes more significant. Figure 10 shows the calculated amplitudes of non-spherical oscillation modes for three bubbles in Figs. 3–6. The distortion amplitudes of bubbles away from glass 1 [Figs. 10(a1), 10(b1), and 10(c1)] are simulated by Eqs. (2) and (8). Figures 10(a2), 10(b2), and 10(c2) display the distortion amplitudes of bubbles attached on the glass plate based on Eqs. (2) and (10). Following the approach of Versluis et al.,29 the initial perturbation an(0) is arbitrarily set to 1 nm, which is much smaller than the size of the bubbles used in the experiment and appears as a reasonable value of radial fluctuations. From Figs. 10(a1) and 10(a2), it now follows that the nonspherical oscillations of the 43 lm bubble can hardly be excited at 6.9 kPa, which holds for the situation before and after its arrival on glass 1. The amplitudes of all modes n ¼ 2 to 6 decay within a few ultrasonic driving cycles, and the higher modes disappear more rapidly (where the least stable mode is n ¼ 3). This outcome coincides with the observed stable spherical bubble oscillations as seen in Fig. 3. The neighborhood of a boundary is not only shifting a bubble’s “breathing mode” (n ¼ 0) resonance, but also the resonances (instability regions) of the higher surface modes n > ¼ 2 to lower frequencies or lower bubble radii, respectively. This can lead to the excitation of otherwise decaying non-spherical oscillations if the bubble approaches the boundary. In our case, this indeed occurs at the higher driving pressure amplitudes. At 12 kPa the n ¼ 3 mode starts to grow only when the bubble arrives on glass 1 [Fig. 10(b2)], which does not happen in the bulk liquid [Fig. 10(b1)]. The other mode amplitudes are still decaying at this driving pressure. Of course, the growth of an unstable mode does not saturate in the model, as it is based upon a linear ordinary differential equation. On the other hand, it is experimentally found that bubbles can oscillate at moderate pressure
FIG. 10. (Color online) Normalized bubble non-spherical pulsation amplitudes for a bubble of 43 lm radius. The mode number is selected from n ¼ 2–6. The non-spherical oscillation amplitudes are displayed for the bubble pulsating at a separation distance dwall ¼ 0.3 mm at 6.9 kPa (a1), 12 kPa (b1), and 17.2 kPa (c1). When the bubble is pulsating on glass 1, its non-spherical oscillation amplitudes are shown for 6.9 kPa (a2), 12 kPa (b2), and 17.2 kPa (c2).
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Xi et al.: Study of non-spherical bubble oscillations
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amplitudes non-spherically without breaking up. Such a phenomenon needs the inclusion of further effects such as nonlinear damping terms or mode-coupling to higher modes. However, this is beyond the scope of this consideration, and we are only concerned here with the small amplitude (linear) stability of the modes in the presence of a wall. Three possibilities that can be responsible for the observed non-spherical oscillation are recognized here. First, the bubble’s translation in the standing wave leads to a higher pressure amplitude at other positions in the acoustic field, in particular at the glass surface (see Fig. 1). The pressure amplitudes are calculated based on a one-dimensional transducer matrix model which were validated for the propose test cell in previous studies.34,35 In the case of Fig. 3, for example, the experimental pressure amplitude increases from 11.3 kPa at dwall ¼ 0.3 mm (the bubble-wall separation distance defined in Fig. 2) to 12 kPa on the glass surface. Second, the pressure threshold for exciting a bubble’s non-spherical oscillation mode is shifted to a lower level due to the wall boundary. This change of amplitude is exemplified in Fig. 11(a), which demonstrates the changes of pressure threshold to excite the modes n ¼ 2 to 5. The solid lines in Fig. 11(a) represent the pressure threshold of a free bubble (no resonance shift is considered for the bubble), while the dashed lines are the thresholds for a bubble attached on the wall (resonance frequency is shifted downward for the bubble). At the driving frequency, the pressure threshold for the mode n ¼ 3 decreases from 21 kPa (free bubble, solid line) to 5 kPa (bubble attached on a wall, dashed line), while the n ¼ 2 mode drops from 16 to 4 kPa. Figure 5 provides a good example of the influence of shifted resonance on the excitation of bubble non-spherical oscillation. Supposing the glass plate is removed from the system and the pressure profile does not change significantly, it can be seen from Fig. 11(a) that the bubble can maintain its spherical shape at 12 kPa which is lower than the threshold of n ¼ 2 and 3 modes (solid lines). On the other hand, in the presence of the wall, the bubble can experience non-spherical oscillation at the same location at the same amplitude due to the lower threshold of n ¼ 2 and 3 modes. This is in good agreement with the observed non-spherical oscillation amplitudes in Fig. 5. This effect is also demonstrated in Fig. 10(b2). It is worth mentioning that the nearby wall can also inhibit the growth of non-spherical oscillation at low amplitude. The bubble surface movement can be constrained when it oscillates on the wall as seen in Fig. 3. Third, the symmetric flow around a free bubble cannot be perfectly established any more for a bubble on the wall. Thus, the presence of glass 1 causes permanent nonspherical perturbations of the bubble wall which might establish growing surface modes. Another factor that can influence the bubble oscillation is the presence of neighboring bubbles. The coalescence of two bubbles creates a new large bubble with a larger radius R0. The relationship between bubble radii and pressure thresholds is shown in Fig. 11(b). The n ¼ 3 mode, for example, could be excited for the 43 lm bubble when it is alone, while a more complicated combination of modes n ¼ 4 to 6 (superposition of several modes) can be triggered when the J. Acoust. Soc. Am., Vol. 135, No. 4, April 2014
FIG. 11. (Color online) Pressure threshold for exciting non-spherical oscillation modes. (a) The thresholds for a bubble of radius of 43 lm as seen in Fig. 5. The pressure thresholds are calculated based on Eq. (3). Solid lines are pressure thresholds for a free bubble, while the dashed line is the pressure threshold for a bubble attached on the wall. The non-spherical modes are numbered from 2 to 5. (b) The thresholds for bubble radii ranging from 20 to 100 lm. The non-spherical modes are numbered from 2 to 5. The pressure thresholds are calculated based on Eq. (3). Solid lines represent pressure threshold of a free bubble, and the scattered lines are thresholds of a bubble attached to a wall.
bubble size grows to 60 lm after the coalescence. Moreover, a large perturbation on the bubble surface is seen during the bubble coalescence [Figs. 6(c) and 6(d)]. This could also contribute to the observed surface modes. C. Bubble translation with a non-spherical shape
In the sound field away from the glass surface, a bubble can oscillate with a non-spherical mode when its size or the external pressure amplitude exceeds a certain threshold. For a bubble of 54 lm radius (Fig. 7), the n ¼ 3 mode can be excited at 6.9 kPa as seen in Fig. 12(a1). The dominating n ¼ 3 mode can be maintained during the bubble’s translation. After arriving on the surface, the bubble experiences a transition period when its spherical oscillation is resumed from the n ¼ 3 mode. At the low pressure amplitude, the wall can temporarily inhibit the non-spherical oscillation. However, the presence of the wall limits the bubble surface movement. The spherical oscillation only Xi et al.: Study of non-spherical bubble oscillations
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FIG. 12. (Color online) Normalized bubble non-spherical pulsation amplitudes for a bubble of 54 lm radius. The mode number is selected from n ¼ 2 to 6. The non-spherical oscillation amplitudes are displayed for the bubble pulsating at 6.9 kPa (a1), 12 kPa (b1), and 17.2 kPa (c1). When the bubble is pulsating on glass 1, its non-spherical oscillation amplitudes are shown for 6.9 kPa (a2), 12 kPa (b2), and 17.2 kPa (c2).
lasts for a short period of time before the bubble starts the non-spherical oscillation again. It can be seen in Fig. 12(a2) that the n ¼ 3 mode experiences a growth as a function of time which is confirmed by the experimental observation. It is worth mentioning that the influence of the wall on the transition between spherical and non-spherical oscillations is only discernable at low pressure amplitude. For the bubble oscillating at a higher amplitude (Fig. 8), the wall can hardly restore the spherical oscillation when the bubble arrives on the wall surface. At a higher amplitude, the n ¼ 3 mode is still dominating when the bubble is away from the wall [Fig. 12(b1)]. Similar to the 6.9 kPa case, all spherical modes are triggered after the arrival of bubble on the surface. At 17.2 kPa, the n ¼ 3 mode is seen to be slightly larger than the other modes which, however, also increases with an increase of time instead of decreasing as seen in Figs. 12(a1) and 12(b1). The growth of all non-spherical modes are seen for all three amplitudes when the bubbles arrive on the surface. It is noticed from Figs. 12(a2), 12(b), and 12(c2) that the simulated mode numbers n ¼ 2 and 3 can be excited for the 54 lm radius bubble at the three amplitudes. With an increase of driving pressure amplitude, more higher modes can be triggered which complicated the bubble oscillation near the wall. VI. CONCLUSION
The excitation of non-spherical bubble dynamics near a surface in an acoustic standing wave field was investigated. High-speed recordings from an experiment were compared to numerical calculations based on the modified RayleighPlesset models. The excitation of non-spherical modes 1740
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requires the pressure amplitude to exceed a certain threshold for a given size of bubble. On the other hand, at a fixed pressure amplitude, different bubble sizes result in different oscillation modes. The neighborhood of a boundary is shifting a bubble’s linear (mode n ¼ 0) resonance, as well as the resonances (instability regions) of the higher surface modes n 2, to lower frequencies or lower bubble radii, respectively. Thus, for a bubble translating with a spherical shape, a nearby surface can lower the pressure threshold of non-spherical oscillation modes. Therefore, the bubble can shift from the spherical oscillation to a non-spherical one just by approaching a wall. The inverse effect is also possible, and the transition between non-spherical and spherical and non-spherical oscillations was observed in the experiment. An increase of pressure amplitude forced the experimental bubbles to undergo stronger surface shape deformations which were substantially enhanced by a neighboring surface. When a bubble approached the target surface, a shifted non-spherical oscillation pressure threshold can force the bubble to switch to another oscillation mode. The bubble oscillation mode is further complicated at a higher pressure amplitude, which can excite additional higher oscillation modes. All of the observed bubble oscillation mode transitions are following the same trend as the theoretical predictions. Understanding the bubble surface deformation near a wall is very important, because it influences the flow field around the bubble. The disturbed flow field is useful in many therapeutic and industrial applications using microbubbles, such as drug delivery, sonoporation, and ultrasonic cleaning, etc. The characteristics of the flow field will be investigated in the future. ACKNOWLEDGMENTS
X.X. and F.C. would like to acknowledge financial support from Lam Research AG (SEZ-Strasse 1, 9050 Villach, Xi et al.: Study of non-spherical bubble oscillations
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Austria). R.M. gratefully acknowledges financial support by the Austrian Federal Ministry of Economy, Family and Youth and the Austrian National Foundation for Research, Technology and Development. 1
D. Krefting, R. Mettin, and W. Lauterborn, “High-speed observation of acoustic cavitation erosion in multibubble systems,” Ultrason. Sonochem. 11, 119–123 (2004). 2 C. Ohl, M. Arora, R. Dijkink, V. Janve, and D. Lohse, “Surface cleaning from laser induced cavitation bubbles,” Appl. Phys. Lett. 89, 074102 (2006). 3 I. Akhatov, O. Lindau, A. Topolnikov, R. Mettin, N. Vakhitova, and W. Lauterborn, “Collapse and rebound of a laser-induced cavitation bubble,” Phys. Fluids 13, 2805–2819 (2001). 4 W. Lauterborn and H. Bolle, “Experimental investigations of cavitationbubble collapse in the neighborhood of a solid boundary,” J. Fluid Mech. 72, 391–399 (1975). 5 C. Ohl, T. Kurz, R. Geisler, O. Lindau, and W. Lauterborn, “Bubble dynamics, shock waves and sonoluminescence,” Philos. Trans. R. Soc., A 357, 269–294 (1999). 6 K. Suslick, “The chemical effects of ultrasound,” Sci. Am. 260, 80–86 (1989). 7 X. Xi, F. Yang, D. Chen, Y. Luo, D. Zhang, N. Gu, and J. Wu, “A targeting drug-delivery model via interactions among cells and liposomes under ultrasonic excitation,” Phys. Med. Biol. 53, 3251–3265 (2008). 8 C. Ohl, M. Arora, R. Ikink, N. De Jong, M. Versluis, M. Delius, and D. Lohse, “Sonoporation from jetting cavitation bubbles,” Biophys. J. 91, 4285–4295 (2006). 9 P. Prentice, A. Cuschieri, K. Dholakia, M. Prausnitz, and P. Campbell, “Membrane disruption by optically controlled microbubble cavitation,” Nat. Phys. 1, 107–110 (2005). 10 A. van Wamel, K. Kooiman, M. Harteveld, M. Emmer, F. ten Cate, M. Versluis, and N. de Jong, “Vibrating microbubbles poking individual cells: Drug transfer into cells via sonoporation,” J. Controlled Release 112, 149–155 (2006). 11 W. Kim, T. Kim, J. Choi, and H. Kim, “Mechanism of particle removal by megasonic waves,” Appl. Phys. Lett. 94, 081908 (2009). 12 A. Thiemann, T. Nowak, R. Mettin, F. Holsteyns, and A. Lippert, “Characterization of an acoustic cavitation bubble structure at 230 kHz,” Ultrason. Sonochem. 18, 595–600 (2011). 13 P. Marmottant and S. Hilgenfeldt, “Controlled vesicle deformation and lysis by single oscillating bubbles,” Nature 423, 153–156 (2003). 14 S. A. Suslov, A. Ooi, and R. Manasseh, “Nonlinear dynamic behavior of microscopic bubbles near a rigid wall,” Phys. Rev. E 85, 066309 (2012). 15 M. Strasberg, “The pulsation frequency of nonspherical gas bubbles in liquids,” J. Acoust. Soc. Am. 25, 536–537 (1953). 16 E. Payne, S. Illesinghe, A. Ooi, and R. Manasseh, “Symmetric mode resonance of bubbles attached to a rigid boundary,” J. Acoust. Soc. Am. 118, 2841–2849 (2005). 17 A. Doinikov, L. Aired, and A. Bouakaz, “Dynamics of a contrast agent microbubble attached to an elastic wall,” IEEE Trans. Med. Imaging 31, 654–662 (2012). 18 A. Doinikov, L. Aired, and A. Bouakaz, “Acoustic scattering from a contrast agent microbubble near an elastic wall of finite thickness,” Phys. Med. Biol. 56, 6951–6967 (2011). 19 A. Doinikov, L. Aired, and A. Bouakaz, “Acoustic response from a bubble pulsating near a fluid layer of finite density and thickness,” J. Acoust. Soc. Am. 129, 616–621 (2011).
J. Acoust. Soc. Am., Vol. 135, No. 4, April 2014
20
V. Garbin, D. Cojoc, E. Ferrari, E. Di Fabrizio, M. Overvelde, S. van Der Meer, N. de Jong, D. Lohse, and M. Versluis, “Changes in microbubble dynamics near a boundary revealed by combined optical micromanipulation and high-speed imaging,” Appl. Phys. Lett. 90, 114103 (2007). 21 M. Plesset, “On the stability of fluid flows with spherical symmetry,” J. Appl. Phys. 25, 96–98 (1954). 22 A. Prosperetti and Y. Hao, “Modelling of spherical gas bubble oscillations and sonoluminescence,” Philos. Trans. R. Soc., A 357, 203–223 (1999). 23 S. Hilgenfeldt, D. Lohse, and M. Brenner, “Phase diagrams for sonoluminescing bubbles,” Phys. Fluids 8, 2808–2826 (1996). 24 J. Loughran, R. Eckersley, and M. Tang, “Modeling non-spherical oscillations and stability of acoustically driven shelled microbubbles,” J. Acoust. Soc. Am. 131, 4349–4357 (2012). 25 A. Maksimov and T. Leighton, “Pattern formation on the surface of a bubble driven by an acoustic field,” Philos. Trans. R. Soc., A 468, 57–75 (2012). 26 S. Zhao, K. Ferrara, and P. Dayton, “Asymmetric oscillation of adherent targeted ultrasound contrast agents,” Appl. Phys. Lett. 87, 134103 (2005). 27 C. Caskey, D. Kruse, P. Dayton, T. Kitano, and K. Ferrara, “Microbubble oscillation in tubes with diameters of 12, 25, and 195 microns,” Appl. Phys. Lett. 88, 033902 (2006). 28 H. Vos, B. Dollet, M. Versluis, and N. de Jong, “Nonspherical shape oscillations of coated microbubbles in contact with a wall,” Ultrasound Med. Biol. 37, 935–948 (2011). 29 M. Versluis, D. Goertz, P. Palanchon, I. Heitman, S. van der Meer, B. Dollet, N. de Jong, and D. Lohse, “Microbubble shape oscillations excited through ultrasonic parametric driving,” Phys. Rev. E 82, 026321 (2010). 30 M. Longuet-Higgins, “Monopole emission of sound by asymmetric bubble oscillations. Part 1. Normal modes,” J. Fluid Mech. 201, 525–541 (1989). 31 A. Doinikov, “Translational motion of a bubble undergoing shape oscillations,” J. Fluid Mech. 501, 1–24 (2004). 32 P. Wilcox, R. Monkhouse, P. Cawley, M. Lowe, and B. Auld, “Development of a computer model for an ultrasonic polymer film transducer system,” NDT&E Int. 31, 51–64 (1998). 33 M. Hill, Y. Shen, and J. Hawkes, “Modelling of layered resonators for ultrasonic separation,” Ultrasonics 40, 385–392 (2002). 34 X. Xi, F. Cegla, M. Lowe, A. Thiemann, T. Nowak, R. Mettin, F. Holsteyns, and A. Lippert, “Study on the bubble transport mechanism in an acoustic standing wave field,” Ultrasonics 51, 1014–1025 (2011). 35 X. Xi, F. Cegla, R. Mettin, F. Holsteyns, and A. Lippert, “Collective bubble dynamics near a surface in a weak acoustic standing wave field,” J. Acoust. Soc. Am. 132, 37–47 (2012). 36 A. Francescutto and R. Nabergoj, “Pulsation amplitude threshold for surface waves on oscillating bubbles,” Acustica 41, 215–220 (1978). 37 P. Birkin, D. Offin, C. Vian, T. Leighton, and A. Maksimov, “Investigation of noninertial cavitation produced by an ultrasonic horn,” J. Acoust. Soc. Am. 130, 3297–3308 (2011). 38 M. A. Ainslie and T. G. Leighton, “Review of scattering and extinction cross-sections, damping factors, and resonance frequencies of a spherical gas bubble,” J. Acoust. Soc. Am. 130, 3184–3208 (2011). 39 Y. A. Kobelev and L. A. Ostrovskii, “Acousto-electrostatic analogy and gas bubbles interaction in liquid,” Sov. Phys. Acoust. 30, 715–716 (1984). 40 H. Oguz and A. Prosperetti, “Bubble oscillations in the vicinity of a nearly plane free surface,” J. Acoust. Soc. Am. 87, 2085–2092 (1990). 41 C. Pozrikidis, Boundary Integral and Singularity Methods for Linearized Viscous Flow (Cambridge University Press, Cambridge, 1992), Chap. 7. 42 A. Doinikov, “Translational motion of a spherical bubble in an acoustic standing wave of high intensity,” Phys. Fluids 14, 1420–1425 (2002). 43 R. Manasseh and A. Ooi, “Frequencies of acoustically interacting bubbles,” Bubble Sci., Eng., Technol. 1, 58–74 (2009).
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