Classification of the mechanisms of wave energy dissipation in the nonlinear oscillations of coated and uncoated bubbles
A.J. Sojahrood, H. Haghi, N.R. Shirazi, R. Karshafian, M.C. Kolios
CClassification of the mechanisms of waveenergy dissipation in the nonlinear oscillationsof coated and uncoated bubbles
A.J. Sojahrood , a , b 1 , H. Haghi , a , b R. Karshfian , a , b and M. C. Kolios , a , b a Department of Physics, Ryerson University, Toronto, Canada b Institute for Biomedical Engineering, Science and Technology (IBEST) apartnership between Ryerson University and St. Mike’s Hospital, Toronto, Ontario,Canada
Abstract
Acoustic waves are dissipated when they pass through bubbly media. Dissipationby bubbles takes place through thermal damping (Td), radiation damping (Rd)and damping due to the friction of the liquid (Ld) and friction of the coating (Cd).Knowledge of the contributions of the Td, Rd, Ld and Cd during nonlinear bubbleoscillations will help in optimizing bubble and ultrasound exposure parameters for therelevant applications by maximizing a desirable parameter. In this work we investigatethe mechanisms of dissipation in bubble oscillations and their contribution to thetotal damping ( W total ) in various nonlinear regimes. By using bifurcation analysis,we have classified nonlinear dynamics of bubbles that are sonicated with their 3rdsuperharmonic (SuH) and 2nd SuH resonance frequency ( f r ), pressure dependentresonance frequency ( P Df r ), f r , subharmonic (SH) resonance ( f sh = 2 f r ), pressuredependent SH resonance ( P Df sh ) and 1 / W total , scattering to dissipation ratio (STDR), maximum wallvelocity and maximum back-scattered pressure from non-destructive oscillations ofbubbles were calculated and analyzed using the bifurcation diagrams. We classifieddifferent regimes of dissipation and provided parameter regions in which a particularparameter of interest (e.g. Rd) can be enhanced. Afterwards enhanced bubble activityis linked to some relevant applications in ultrasound. This paper represents the firstcomprehensive analysis of the nonlinear oscillations regimes and the correspondingdamping mechanisms. Email: [email protected]
Preprint submitted to Elsevier 17 September 2020 a r X i v : . [ phy s i c s . a pp - ph ] S e p Introduction
An ultrasonically excited bubble is a highly nonlinear oscillator in which de-terministic chaos manifests itself [1, 2, 3]. When a high pressure acoustic fieldis generated in an aqueous medium, the rare faction cycle may exceed theattractive forces among liquid molecules generating cavitation bubbles. Bubblesbegin oscillating and emit sound [4, 5, 6]. The spectral components of theemitted sound consist of harmonics and subharmonics of the incident soundwave center frequency and broadband noise (Lauterborn & Holzfuss 1991 [3]).The nonlinear frequency content of the emitted sound by bubbles has foundits applications in contrast enhanced diagnostic ultrasound to visualize thevascular structure [7, 8, 9] with superior contrast. Bubbles signatures are alsoused for monitoring treatments in therapeutic ultrasound [10, 11, 12].The pressure emitted by collapsing bubbles may form a shock wave (Radek1972; Vogel et al. 1986) [5, 13], that can mechanically damage nearby nearbystructures. Bubble oscillations generate micro-streaming in the liquid whichresults in shear stresses on the objects in its vicinity and micro-mixing in theliquid [14, 15]. The induced shear stresses and the emitted shock-waves hasfound their applications in industry (cleaning the micro-structures [14, 15, 16])and medicine (e.g. enhanced drug and gene delivery [17, 18, 19], blood brainbarrier opening [20, 21] and shock wave lithotripsy and histotripsy [22, 23]).Ultrasonically excited bubbles can focus and concentrate the acoustic energyfrom the macro-scale (acoustic wave) to the micro-scale and nano-scale [19, 24]generating extremely high temperatures and pressures as the bubbles collapse.This leads to molecular disassociation which triggers the production of highlyreactive free radicals [24, 25, 26] which then interact with other substances inthe solution. This phenomenon has been shown useful in numerous industrialprocessing applications ranging from sonochemistry [24, 25, 26] (chemicalreaction rate enhancement and treatment of organic compounds) to the foodindustry [27] and medicine (sonodynamic therapy [28]). Bubbles can focus andamplify the energy of the sound field by more than 11 orders of magnitude,which is sufficient not only to break chemical bonds but also to induce lumi-nescence [29]. Local sound amplification and enhanced dissipation of acousticenergy by bubbles been used to enhance the heating generated by ultrasoundduring ultrasound thermal therapies and high intensity focused ultrasound(HIFU) tumor ablation [30].Understanding and enhancing a specific type of bubble oscillatory pattern canhelp in enhancing the outcome of the relevant application. For example, incontrast enhanced ultrasound the goal is maximizing the radiated pressureby the bubbles while keeping the dissipation of energy due to bubble atten-uation minimum [31, 32, 33]. This will lead to enhanced contrast and bettervisualization of the target and eliminating the shadowing in ultrasonic images[31, 32, 33]. Shadowing [34, 35] is caused by the dissipation of the ultrasonicenergy by bubbles which leads to a weaker signal intensity from underlying2issue. In HIFU the goal is to increase the dissipation at the focus while re-ducing pre-focal shielding and energy dissipation by bubble oscillations. Here,knowledge of the pressure dependent dissipation effects and the advantage ofthe sharp pressure gradients of HIFU transducers facilitate the desired effect[32, 33, 36].Bubbles dissipate the acoustic energy through radiation damping (Rd), thermaldamping (Td), damping due to the viscosity of the liquid (Ld) and damping dueto the friction of the coating (Cd) [37, 38, 39, 40, 41]. Despite the importance ofdetailed knowledge of the energy dissipation mechanisms in bubble oscillations,the majority of previous studies have been limited by linear approximations[37, 42, 43, 44, 45]. Linear studies simplify the bubble oscillations to verysmall amplitudes at low excitation pressures (e.g. 1 kPa) [37, 42]. However,bubble oscillations are nonlinear and energy dissipation depends highly on theexcitation pressure [39, 40, 41, 46]. Moreover, the majority of the applicationsare based on sending ultrasound pulses of high pressure amplitude; thus, linearapproximations are inappropriate to model bubble oscillations.Despite the importance of the knowledge on nonlinear energy dissipation bybubbles; however, there are only few recent studies that explored the pressuredependent effects on energy dissipation [38, 39, 40, 41, 46, 47]. Louisnard [38]derived the pressure dependent energy equations by considering the conserva-tion of mass and momentum in a bubbly media and used the Rayleigh-Plessetequation for bubble oscillations [48]. He derived the dissipation equations forLd and Td. His analysis showed that energy dissipation is pressure dependentand predictions of the linear model can be orders of magnitude smaller thanthe pressure dependent model. Jamshidi & Brenner used Louisnard’s approachand Keller-Miksis equation [49] to incorporate the compressibility effects up tothe first order of Mach number. They were able to derive Ld, Td and Rd. Theiranalysis showed that Rd has an important role in energy dissipation and as istypically done cannot be neglected [39]. In our recent work, we showed thatequations derived by Jamshidi & Brenner need to be corrected as their modelpredicts non-physical values for Rd near resonance and predictions of Rd arenot consistent with the predictions of the scattered pressured energy (Sd) bybubbles [40]. We presented the corrected forms of Ld, Rd and Td. We showedthat dissipation terms are highly pressure dependent and as pressure increasesRd may grow faster than Td and Ld; thus, there exist optimum pressure andfrequency ranges where the scattering to dissipation ratio (STDR) can bemaximized [40, 41]. Moreover, we showed that the STDR which can be usedas standardization parameter to assess the efficacy of bubble oscillations [40]in applications is pressure dependent. STDR should be used in conjunctionwith Rd and the maximum scattered (re-radiated) pressure by bubbles for amore complete assessment of a given control parameter for bubble oscillationoptimization [40].Using the same approach as in [40], we derived the nonlinear energy dissipationequations for a coated bubble [41]. We analyzed the resonance power curves forfree and encapsulated bubbles and showed that Td can be neglected for coated3ubbles that have C3F8-like gas cores. We also showed that although Td isthe dominant dissipation mechanism for large uncoated bubbles; at higherpressures Rd can supersede Td. Moreover, Cd is the strongest dissipationmechanism in the oscillations of the coated bubbles; pressure increase however,there are instances in which Rd is stronger than Ld and Td.In this paper we provide a detailed analysis of the pressure dependent dissi-pation mechanisms by bubble oscillations and role of each of the dissipationcomponents (Td, Ld,Rd and Cd) at various nonlinear regimes. Knowledge ofthe pressure dependent dissipation effects and the examination of each con-tributing component will help us better understand bubble related phenomenaand enhance a desirable effect in bubble oscillations.In this paper we have classified major nonlinear regimes of the oscillations forfree and coated bubbles. In this regard, our recent comprehensive approachis used to analyze the bubble oscillations [50] as a function of pressure. Themajor nonlinear regimes that are considered here are 2nd and 3rd SuH resonantoscillations, 3 /
2, 5 / / f r ), pressure dependent resonance( P Df r ),1 /
2, 1 / W total ) can be maximized or minimized. These findings are then related tosome of the current applications of bubbles. The dynamics of a coated bubble oscillator including compressibility effectsto the first order of Mach-number can be modeled using the Keller-Miksis-Church-Hoff (KMCH) model [41, 42, 49]: ρ (cid:34)(cid:32) − ˙ Rc (cid:33) R ¨ R + 3 / R (cid:32) − ˙ R c (cid:33)(cid:35) = (cid:32) Rc + Rc ddt (cid:33) (cid:32) P g − µ L ˙ RR − µ sh (cid:15)R ˙ RR − G s (cid:15)R (cid:18) R − R R (cid:19) − P − P (cid:33) (1)4 hermal parameters of the gases at 1 atmGas type L( WmK ) c p ( kJkgK ) c v ( kJkgK ) ρ g ( kgm )Air [53] 0 . C × T C=5 . × WmK . Where ρ and c are respectively the density and sound speed of the medium,R is the radius at time t, ˙ R is the bubble wall velocity, ¨ R is the bubble wallacceleration, R is the initial radius of the bubble, µ and µ sh are the viscosityof the liquid and shell (coating) respectively, (cid:15) is the thickness of the coating, G s is the shell shear modulus, P g is the gas pressure inside the bubble, P isthe atmospheric pressure (101.325 kPa) and P is the acoustic pressure givenby P = P a sin (2 πf t ) with P a and f are respectively the excitation pressureand frequency. In this paper for all of the simulations of the coated bubbles G s =50 MPa and µ sh = . R ( µm ) − . θ ( nm ) [51] with θ = 4 nm . The gas inside thebubble was chosen to be C3F8 and the surrounding medium water. The dynamics of the uncoated bubble including the compressibility effectsto the first order of Mach number can be modeled using Keller-Miksis (KM)equation[49]: ρ [(1 − ˙ Rc ) R ¨ R + 3 / R (1 − ˙ R c )] = (1 + ˙ Rc )( G ) + Rc ddt ( G ) (2)where G = P g − µ L ˙ RR − σR − P − P A sin (2 πf t ).In this equation, R is radius at time t, R is the initial bubble radius, ˙ R is thewall velocity of the bubble, ¨ R is the wall acceleration, ρ is the liquid density(998 kgm ), c is the sound speed (1481 m/s), P g is the gas pressure, σ is thesurface tension (0.0725 Nm ), µ is the liquid viscosity (0.001 Pa.s), and P A and f are the amplitude and frequency of the applied acoustic pressure. The valuesin the parentheses are for pure water at 293 K. In this paper the gas insidethe uncoated bubble is air and water is the host media.5 .3 Thermal effects If thermal effects are considered, P g is given by Eq. 5 [49, 50, 51, 52, 53]: P g = N g KT πR ( t ) − N g B (3)Where N g is the total number of the gas molecules, K is the Boltzman constantand B is the molecular co-volume of the gas inside the bubble. The averagetemperature inside the gas can be calculated using Eq. 6 [49]:˙ T = 4 πR ( t ) C v (cid:32) L ( T − T ) L th − ˙ RP g (cid:33) (4)where C v is the heat capacity at constant volume, T =293K is the initial gastemperature, L th is the thickness of the thermal boundary layer. L th is givenby L th = min ( (cid:114) aR ( t ) | ˙ R ( t ) | , R ( t ) π ) where a is the thermal diffusivity of the gas whichcan be calculated using a = Lc p ρ g where L is the gas thermal conductivity and c p is specific heat capacity at constant pressure and ρ g is the gas density.Predictions of the full thermal model have been shown to be in good agreementwith predictions of the models that incorporate the thermal effects using thePDEs [55] that incorporate the temperature gradients within the bubble. Tocalculate the radial oscillations of the coated bubble and uncoated bubblewhile including the thermal effects Eqs. 1 and Eq. 2 are respectively coupledwith Eq. 3 and 4 and then solved using the ode45 solver of Matlab.6 .4 Nonlinear terms of dissipation for the KMCH model We have derived the equations for the average power loss in the oscillations ofthe KMCH model [41]:
T d = − πT (cid:90) T R ˙ RP g dtLd = 16 πµ L T (cid:90) T R ˙ R dtCd = 48 πµ sh εR T (cid:90) T ˙ R R dtGd = 48 πG s εR T (cid:90) T (cid:32) ˙ RR − R ˙ RR (cid:33) dtRd = 1 T (cid:90) T (cid:32) π (cid:34) R ˙ R c ( P − P g ) + R ˙ Rc (cid:16) ˙ P − ˙ P g (cid:17) + 4 µ L R ˙ R ¨ Rc +12 µ sh εR (cid:32) ˙ R ¨ RcR − R cR (cid:33) + 12 G s εR (cid:32) − R cR + 3 R ˙ R cR (cid:33)(cid:35) − ρR ˙ R c − ρR ˙ R ¨ Rc (cid:33) dt (5)Where Td, Ld, Cd, Rd and Gd are the average dissipated powers due tothermal, Liquid viscosity , coating viscosity, re-radiation and stiffness of thecoating. In simulations we did not present the values for Gd since it is alwayszero for a full cycle. T is the integration time and can be given as n/f wheren=1,2...... . In this paper the integrals are performed over the last 20 cycles ofa 500 cycles pulses to avoid the transient bubble behavior.7 .5 Nonlinear terms of dissipation for the KM model We have derived the dissipation power terms of the KM model as follows [40]:
T d = − T (cid:90) T ( P g ) ∂V∂t dtLd = 16 πµ L T (cid:90) T (cid:16) R ˙ R (cid:17) dtRd = 1 T (cid:90) T (cid:20) πc (cid:16) R ˙ R (cid:16) ˙ RP + R ˙ P − / ρ ˙ R − ρR ˙ R ¨ R (cid:17)(cid:17) − (cid:32) ˙ Rc P g + Rc ˙ P g (cid:33) ∂V∂t + 16 πµ L R ˙ R ¨ Rc (cid:35) dt (6)All the dissipated powers were calculated for the last 20 cycles of pulses with500 cycles length. Simulations were carried out in Matlab using ODE45 with thehighest possible relative and absolute tolerance. The time steps for integrationin each simulation were ≤ − f . Bifurcation diagrams are valuable tools to analyze the dynamics of nonlinearsystems where the qualitative and quantitative changes of the dynamics ofthe system can be investigated effectively over a wide range of the control pa-rameters. In this paper, we employ a more comprehensive bifurcation analysismethod introduced in [50, 56].
When dealing with systems responding to a driving force, to generate thepoints in the bifurcation diagrams vs. the control parameter, one option isto sample the R(t) curves using a specific point in each driving period. Theapproach can be summarized by: P ≡ ( R (Θ)) { ( R ( t ) , ˙ R ( t )) : Θ = nf } where n = 480 , ...
500 (7)Where P denotes the points in the bifurcation diagram, R and ˙ R are the timedependent radius and wall velocity of the bubble at a given set of controlparameters of ( R , P , P A , c , k , µ , G s , µ sh , θ , σ , f ) and Θ is given by nf . Points8n the bifurcation diagram are constructed by plotting the solution of R ( t ) attime points that are multiples of the driving acoustic period. The results areplotted for n = 480 −
500 to ensure a steady state solution has been reached.
As a more general method, bifurcation points can be constructed by settingone of the phase space coordinates to zero: Q ≡ max ( R ) { ( R, ˙ R ) : ˙ R = 0 } (8)In this method, the steady state solution of the radial oscillations for eachcontrol parameter is considered. The maxima of the radial peaks ( ˙ R = 0) areidentified (determined within n = 480 −
500 cycles of the stable oscillations)and are plotted versus the given control parameter in the bifurcation diagrams.The bifurcation diagrams of the normalized bubble oscillations (
R/R ) arecalculated using both methods a) and b). When the two results are plottedalongside each other, it is easier to uncover more important details about theSuH and UH oscillations, as well as the SH and chaotic oscillations. In this section various nonlinear oscillation regimes of coated and uncoatedbubbles are introduced by visualizing the radial oscillations of the bubble asa function of pressure at various frequencies. Then we build a link betweendifferent nonlinear oscillation regimes and the dissipated powers.In the simulations, the uncoated bubbles that enclose air have initial radii of10 µm and 2 µm . The bubble with R = 10 µm is chosen as it will have strongthermal damping due to its bigger size. The bubble with R = 2 µm is chosenas viscous effects are strong due to its size (Results related to this case arepresented in Appendix).For the coated bubbles we investigated the bubbles with initial radii of 1 and4 µm . The bubble with R = 4 µm is probably the largest bubble that can beused in medical applications (as the capillaries have diameters around 8 µm [8]). Therefore, this bubble has the highest possible size dependent Td (Resultsof this case are presented in Appendix). The bubble with R = 1 µm is alsochosen as it is in the typical range of the contrast agents that are used inmedical applications and viscous effects strongly influence its dynamics.9 .1 Bifurcation structure and dissipation mechanisms of uncoated bubbles3.1.1 The case of an uncoated air bubble with R = 10 µm Figure 1 shows the bifurcation structure of the normalized oscillations (
R/R )as a function of acoustic pressure of an uncoated air bubble with R = 10 µm andthe corresponding dissipated powers due to Ld, Td and Rd for (0 . f r ≤ f ≤ . f r (Fig. 1a) an increase in pressure resultsin the generation of 3rd order SuH oscillations at P a (cid:117) kP a (the blue curveshows three maxima for a period one oscillation (1 solution in the red graph)).The red curve undergoes a period doubling (Pd) bifurcation concomitant witha Pd in the blue graph at P a (cid:117) kP a . This results in 7 / P a (cid:117) kP a . With a slight pressureincrease a saddle node bifurcation takes place to P1 oscillations with 3 maximaof higher amplitude. At this point the bubble may not sustain stable oscillationsas R/R > P a ≤ kP a , Rd is the weakest damping mechanismwith Td the strongest mechanism (approximately 2 orders of magnitude larger).Rd grows faster than other damping mechanisms with increasing pressure andat P a (cid:117) kP a concomitant with the appearance of 3rd SuH oscillations, Rdbecomes equal to Ld. Rd becomes stronger than Ld when UH oscillations occur;later, simultaneous with the saddle node bifurcation Rd undergoes a largeincrease and becomes the strongest damping mechanism. Td is the dominantmechanism for pressures below 100 kPa (the saddle node bifurcation) and at ≈
130 kPa
Rd > T d = Ld .When f = 0 . f r (Fig 1c); 2nd order SuH occurs in the oscillations of thebubble at P a (cid:117) kP a ; this manifests itself as a P1 oscillation (1 red line) with2 maxima (two solutions for the blue curve). Radial oscillations grow withincreasing pressure and at P a (cid:117) kP a the red curve undergoes a Pd which iscoincident with a Pd for the blue curve; this results in 5 / P a (cid:117) kP a ); further at P a (cid:117) kP a a giant P1 resonance emerges out ofchaos. Possible bubble destruction occurs at ≈ kP a (black horizontal line( R/R > T d > Ld > Rd . Later, concomitantwith saturation of 2nd order SuH oscillations at ≈ kP a (red line becomesequal to one of the maxima indicating the wall velocity becomes in phase withthe driving acoustic pressure). Rd becomes equal to Td and gets stronger thanLd during UH oscillations. Td is the dominant mechanism at P a < kP a ;however, when UH oscillations are saturated, Rd supersedes Td and stays10 Pressure (Pa) -20 -18 -16 -14 D i ss i p a t e d po w e r ( W ) R0=10 m,f=0.25*fr
TdRdLd (a) (b)
Pressure (Pa) -20 -18 -16 -14 -12 D i ss i p a t e d po w e r ( W ) R0=10 m,f=0.5*fr
TdRdLd (c) (d)
Pressure (Pa) R / R R0=10 m uncoated bubble,f=0.9*fr
Pressure (Pa) -18 -17 -16 -15 -14 -13 D i ss i p a t e d po w e r ( W ) R0=10 m,f=0.9*fr
TdRdLd (e) (f)
Pressure (Pa) R / R R0=10 m,f=1*fr
Pressure (Pa) -17 -16 -15 -14 D i ss i p a t e d po w e r ( W ) R0=10 m,f=1*fr
TdRdLd (g) (h)
Fig. 1. Bifurcation structure (left column) and the dissipated power as a function ofpressure (right column) of the oscillations of an uncoated air bubble with R = 10 µm for f = 0 . f r (a-b)- f = 0 . f r (c-d)- f = 0 . f r (e-f) & f = f r (g-h). f = 0 . f r (Fig. 1e), P1 oscillations (with 1 maxima) undergo a saddlenode bifurcation to P1 oscillations of higher amplitude at P a (cid:117) kP a . Thebubble possibly is destroyed at P a (cid:117) kP a (black horizontal line). Furtherincrease in pressure results in Pd at 175 kPa; P2 oscillations undergo a cascadeof Pds to chaos at 210 kPa. The corresponding dissipated power is presented inFig. 1f. For pressures below the saddle node (SN) bifurcation Td is the strongestdamping mechanism (an order of magnitude larger) with T d > Ld (cid:117) Rd .Concomitant with the SN, (note that at this pressure the wall velocity becomesin phase with the driving pressure) Rd becomes stronger than Ld and at 100kPa it surpasses the initially larger Td. Further increase in pressure results inthe fastest growth rate in Rd and the slowest growth rate in Td. Simultaneouswith Pd and during majority of the P2 oscillation regime, Rd, Ld and Tdstay approximately constant (this can be due the decrease in wall velocityconcomitant with Pd when bubble is sonicated with a frequency near itsresonance frequency [32]). During chaotic oscillations Rd > T d > Ld withfluctuations due to sporadic oscillations.For f = f r (Fig. 1g), at lower pressures ( P a < kP a ) oscillations are P1 andthe wall velocity is in phase with the driving acoustic force (blue and redcurve are on top of each other). Further pressure increases result in possiblebubble destruction at P a = 100 kP a (black horizontal line meets the blue line R/R > P a (cid:117) kP a , Pd occurs and choas appears for 205 < P a < P a < kP a where wall velocity is in phasewith the driving pressure T d > Rd = Ld and there is a very sharp growth forall the damping factors (possibly due to the resonant nature of oscillations).Rd becomes bigger than Ld above 25 kPa and grows faster than both Ld andTd until it becomes equal to Td at at P a (cid:117) kP a . Rd becomes sightly higherthan Td when Pd occurs; however, the occurrence of Pd decreases the rate ofgrowth of the damping powers and they which stays relatively constant duringP2 oscillations (due to possibly the decrease of the wall velocity during P2oscillations when f = f r [32, 58]). Chaotic oscillations result in a slight decreasein Td but Rd keeps growing and at the giant resonance Rd undergoes a largeincrease and becomes approximately two orders of magnitude larger than theother damping factors. Occurrence of the P2 giant resonance is concomitantwith a decrease in Td. The reduction in Td is concomitant with the occurrenceof the giant resonance may lead to better sonochemical efficacy as highertemperatures are created while at the same time thermal conduction becomesmore limited.Figure 2a shows the case of sonication with f = 1 . f r . We have chosen thisfrequency as the bubble is able to undergo non-destructive Pd ( R/R < f = f r most likely results in bubble destruction before the development of any P212 Pressure (Pa) R / R R0=10 m uncoated bubble,f=1.2*fr
Pressure (Pa) -18 -16 -14 D i ss i p a t e d po w e r ( W ) R0=10 m,f=1.2*fr
TdRdLd (a) (b)
Pressure (Pa) -18 -16 -14 D i ss i p a t e d po w e r ( W ) R0=10 m,f=1.75*fr
TdRdLd (c) (d)
Pressure (Pa) -18 -16 D i ss i p a t e d po w e r ( W ) R0=10 m,f=2*fr
TdRdLd (e) (f)
Pressure (Pa) -20 -18 -16 D i ss i p a t e d po w e r ( W ) R0=10 m uncoated bubble,f=3*fr
TdRdLd (g) (h)
Fig. 2. Bifurcation structure (left column) and the dissipated powers (right column)of the oscillations of an uncoated air bubble with R = 10 µm for f = 1 . f r (a-b)- f = 1 . f r (c-d)- f = 2 f r (e-f) & f = 3 f r (g-h). f = f r ). Fig. 2a shows that when f = 1 . f r , radial oscillations are initially of P1 and monotonically increase inamplitude as excitation pressure increases. At P a (cid:117) kP a Pd takes place;P2 oscillations then undergo a SN bifurcation to a P3 oscillations (propertiesof this P3 oscillation has been studied in [58]) which can be concomitantwith bubble destruction as
R/R >
2. The bubble oscillations return to P2after a very small window of chaos. Another chaotic window appears throughsuccessive Pd. A giant P2 resonance emerges out of the chaotic window when P a > kP a which later undergo successive Pds to chaos. The dissipatedpowers are shown in Fig. 2b. Td is the strongest damping factor for pressuresbelow 190 kPa. For P a <
80 kPa,
T d > Rd (cid:117) Ld . Rd becomes stronger than Ldwith increasing pressure above 80 kPa. At P a =190 kPa, Rd becomes strongerthan Td simultaneous with the SN bifurcation for P3 oscillations; however,as soon as P3 converts to P2, Td becomes larger than Rd. Emergence of theP2 giant resonance is simultaneous with a large increase in Rd and Ld and asubsequent decrease in Td. This can be due to the faster collapse with higherwall velocity and acceleration resulting in an increase in Rd and Ld; however,due to the fast collapse there is not enough time for temperature conductionthus Td decreases. In this region Rd is an order of magnitude larger than Ldand Td and its the only region in this pressure range where Ld is strongerthan Td.Fig. 2c displays the case of sonication with f = 1 . f r which is the pressuredependent SH resonance frequency of the bubble ( P Df sh [59]). This frequencyis chosen as the SN bifurcation leads to non-destructive oscillations. Oscillationsare of P1 initially; pressure increase results in Pd at (cid:117) kP a . P2 oscillations(with two maxima) undergo a SN bifurcation to P2 oscillations (with onemaximum) of higher amplitude (cid:117) kP a . At (cid:117) kP a second maximare-emerges with the same amplitude of the smaller solution in the red curve(indicating that wall velocity is in phase with the excitation pressure onceevery two acoustic cycles). At (cid:117) kP a , R/R = 2 (black horizontal line);beyond this pressure the bubble may not sustain non-destructive oscillations.P2 oscillations undergo Pds to a P4 solution which later undergoes successivePds to chaos at P a = 300 kP a . A giant P3 (with two maxima) resonance emergesout of the chaotic window at (cid:117) kP a . Fig. 2d shows that for P a < kP aT d > Rd > Ld . Occurrence of the SN bifurcation (over-saturation of SHsignal [59]) results in a fast increase in Rd and enhancement in the STDR. Rdgrows with pressure increase during the P2 oscillations; however Td and Lddo not increase. Rd, Td and Ld undergo sporadic fluctuations during chaos.Emergence of giant resonance results in a sharp increase in Rd and Ld and asmall decrease in Td. Rd > Ld > T d for the P3 giant resonance oscillationsregime. The decrease in Td and the faster and larger radial collapses indicatethat higher temperatures are generated while the heat conduction becomeslimited. The higher temperatures can have consequences in enhancing chemicalreactions within the bubble.When sonicated with f = 2 f r ( f sh ), oscillations undergo a Pd at P a = 100 kP a ;142 oscillations increase in amplitude and evolve in a shape of a bow-tie[59]. Consistent with previous observations [59] sonication with f sh resultsin the largest pressure range with stable P2 oscillations. At P a (cid:117) kP a asmall window of P6 oscillations appear through a SN bifurcation with eachsolution undergoing Pds to P12 (the properties of this oscillation regimehave been studied in the appendix of (Chapter 4) [59]). Oscillations returnto P2 which then undergo Pd to P4 oscillations. For a small window ofexcitation pressure P12 oscillations appear through a SN bifurcation; however,here because R/R > (cid:117) kP a chaos appears. A P3 giant resonanceemerges out of the chaotic window which later undergo successive Pds tochaotic oscillations. For P a < kP a , T d > Rd > Ld . After the occurrence ofPd, Td remains relatively constant with increasing pressure while Rd growsfaster than Ld as pressure increases. Eventually at P a (cid:117) kP a Rd becomesequal to Td. The occurrence of giant resonance results in a sharp increase inRd and Ld and Rd becomes the strongest damping factor with
Rd > Ld > T d .Regeneration of chaos results in a decrease in Rd and Ld with Rd (cid:117) T d > Ld .When f = 3 f r (Fig. 2g); radial oscillations grow very slowly and monotonicallywith pressure; at P a (cid:117) kP a a SN bifurcation takes place and oscillationsbecome P3 (3 solutions for the red curve with 2 maxima). Properties ofthese oscillations have been studied in [60, 61, 62]. At (cid:117) kP a oscillationsundergo a SN bifurcation to P6 oscillations for small excitation pressure windowwhich then transition to P12 and then back to P3 oscillations. P12 occurs at ≈ kP a through Pds. P12 oscillations then switch to P1 oscillation withpressure increase. Power dissipation curve is shown in Fig. 2h. Here Rd isthe strongest damping mechanism for all the studied pressure ranges with Rd > T d > Ld . SN bifurcation results in a sharp increase in the dissipatedpowers at 300 kPa with Td exhibiting the largest increase. R = 1 µm Fig. 3a shows the dynamics of a C3F8 coated bubble with R = 1 µm when f = 0 . f r . Oscillations are initially P1 with one maximum, later at about P a = 160 kP a , 3 maxima are generated in the bubble oscillations which growin amplitude as pressure increases, undergoing a SN bifurcation to higheramplitude oscillations at about 320 kPa. In this region (P1 with 3 maxima) the3rd harmonic of the backscattered pressure is maximum and the bubble is inthe 3rd order SuH oscillation mode. Pd occurs at about 350 kPa, leading to aP2 signal with 6 maxima and 7 / Pressure (Pa) -25 -20 -15 D i ss i p a t e d po w e r ( W ) R0=1 m coated bubble,f=0.25*fr
TdRdLdCd (a) (b)(c) (d)
Pressure (Pa) -24 -22 -20 -18 D i ss i p a t e d po w e r ( W ) R0=1 m coated bubble,f=0.8*fr
TdRdLdCd (e) (f)
Pressure (Pa) -22 -20 -18 D i ss i p a t e d po w e r ( W ) R0=1 m coated bubble,f=1*fr
TdRdLdCd (g) (h)
Fig. 3. Bifurcation structure (left column) and the dissipated power as a function ofpressure (right column) of the oscillations of a coated C3F8 bubble with R = 1 µm for f = 0 . f r (a-b)- f = 0 . f r (c-d)- f = 0 . f r (e-f) & f = f r (g-h). (cid:117) kP a ; at the same time R/R exceeds 2 thus,the bubble may not sustain long lasting non-destructive oscillations beyond thispressure. At 405 kPa, a P2 oscillation regime with large amplitude emerges outof the chaotic window. Oscillations become chaotic again through successivePds of the P2 signal and the chaotic window extends until ≈ kP a . Abovethis pressure a P2 giant resonance emerges out of chaos, and later undergoessuccessive Pds to chaos. Fig. 3b shows that at pressures lower than 160 kPa(generation of 3 maxima in the osculations) Cd > Ld > T d > Rd . Above 160kPa, Rd increases beyond Td and
Cd > Ld > Rd > T d until SN bifurcationoccurs at about 320 kPa. Rd grows faster and when SN occurs it undergoes asharp increase alongside Ld and Cd.
Cd > Ld = Rd > T d above SN bifurcationand during UH oscillations. Rd exceeds Ld when
R/R > Cd = Rd > Ld > T d until about 900 kPa where Rd slightly exceeds Cd.Emergence of the giant P2 resonance leads to a sharp increase in Rd and adecrease in Td similar to previous cases.When f = 0 . f r (Fig. 3b) P1 oscillations increase in amplitude with pressureand at about 85 kPa a second maxima appear in the bubble oscillations.Oscillations keep growing and at about 300 kPa the red curve becomes equalin amplitude to the highest amplitude maxima (indicating the wall velocity ofone of the maxima becomes in phase with the driving acoustic field). At about320 kPa, Pd occurs and oscillations become P2 with 4 maxima indicatingthe generation of 5 / R/R exceeds 2 the bubble possiblyundergoes destruction. The dissipation curves are shown in Fig. 3d. For lowerpressures Cd > Ld > Rd = T d ; however, due to the faster growth of Rdcompared to other dissipation mechanisms, it supersedes Td at about 40 kPaand becomes equal to Ld when Pd takes place. When the P2 oscillations withhigher amplitude emerge out of the chaotic window, the dissipation powersundergo a sharp increase with
Cd > Rd = Ld > T d .The case of sonication with f = 0 . f r ( P Df r ) is presented in Fig.7e. P1oscillations undergo SN bifurcation to higher amplitude at (cid:117) kP a and atthe same time the value of the red curve becomes equal to the maxima in theblue curve (indicating the wall velocity is in phase with the driving signal).Oscillations grow with pressure increase and at 410 kPa, Pd takes place leadingto P2 oscillations until 500 kPa. Chaos is then generated through successiveperiod doubling bifurcations at 510 kPa. For this frequency Cd > Ld > Rd >T d in the studied pressure range. There is a sharp increase in the dissipationpower when SN takes place. Furthermore concomitant with Pd; Cd, Ld andRd decrease due to reduced wall velocities [32].Fig. 3g displays the case of sonication with f = f r . Initially the value of the redcurve is equal to the oscillation amplitude in the blue curve and above 100 kPathe two curves diverge (this is because f r shifts to P Df r as pressure increases1732] and when f = f r oscillations are only resonant at lower pressures) and Pdtakes place at (cid:117) kP a . P2 oscillations undergo successive Pds to chaos at620 kPa. Chaos stretches beyond 1 MPa with oscillation amplitudes exceeding R/R = 2 at 700 kPa. The dissipated power curves are presented in Fig. 3h.Similar to the case of f = 0 . f r , Cd > Ld > Rd > T d and occurrence of Pdleads to a slight decrease in Cd, Rd and Ld.Fig. 4a displays the case of sonication with f = 1 . f r . The P1 oscillationamplitude increases with pressure and Pd occurs at about 570 kPa. A smallperiod bubbling window takes place for (cid:117) − kP a and initiation of chaosis at about 860 kPa. When chaos is initiated, R/R >
2. The correspondingpower curves in Fig. 4b show that similar to the case of f = 0 . f r and f r , Cd > Ld > Rd > T d for P a < kP a where period bubbling takes place.Occurrence of Pd at 570 kPa is concomitant with a decrease in Cd, Ld andRd and when bubbling occurs Cd (cid:117) Ld (cid:117) Rd > T d . Generation of suddenchaos at ≈ kP a is simultaneous with a sudden increase in Cd, Ld and Rdwith Cd > Ld > Rd > T d right after the onset of chaos. Further increases inpressure result in a faster growth in Rd making Rd (cid:117)
Cd at ≈ f = 1 . f r ( P df sh ). P1 oscillation amplitude grow with pressure increase and at 580 kPa aSN bifurcation from P1 oscillations to P2 oscillations of higher amplitude takesplace. P2 oscillations then grow with pressure increase and undergo further Pds.After a small window of P6-P12 oscillations chaos is generated. At 1.2 MPaoscillation amplitude exceeds 2 and possible bubble destruction may take place.The corresponding power graphs are depicted in Fig. 4d. Cd > Ld (cid:117)
Rd > T d for P a < kP a . Occurrence of the SN results in a sharp increase in Cd, Rd,Ld and Td with Td exhibiting the highest increase.When f = 2 f r (Fig. 4c) P1 oscillations undergo Pd at 400 kPa; P2 oscillationslater evolve in a form of a bow-tie [58] (red curve) undergoing successive Pdto chaos. For P a less than the pressure threshold of Pd, Rd (cid:117) Cd > Ld > T d .When Pd occurs, Cd grows larger than Rd and Rd becomes equal to Ld. Tdexhibits the largest growth when Pd occurs. During P2 oscillations Rd growsfaster than other damping factors exceeding Ld at (cid:117) kP a .When f = 3 f r (Fig. 4g) P1 oscillations undergo a SN bifurcation to P3oscillations of higher amplitude. Pressure increase results in an increase inthe amplitude of the P3 oscillations and at (cid:117) . M P a , Pd takes place andoscillations become P12. Later through multiple Pds a small chaotic windowappears which is followed by a sudden onset of P1 oscillations for the rest ofthe pressures studied here. The corresponding dissipated power graphs areshown in Fig. 4h. Rd is the strongest dissipated power for the studied pressurerange here with
Rd > Cd > Ld > T d . When SN takes place, similar to thecase of R = 4 µm all dissipated powers undergo a sharp increase; however, Tdexhibits the largest growth potentially due to more surface area available forheat transfer. 18 Pressure (Pa) R / R R0=1 m,f=1.2*fr conventional methodmaxima method
Pressure (Pa) -26 -24 -22 -20 -18 D i ss i p a t e d po w e r ( W ) R0=1 m coated bubble,f=1.2*fr
TdRdLdCd (a) (b)
Pressure (Pa) -26 -24 -22 -20 -18 D i ss i p a t e d po w e r ( W ) R0=1 m coated bubble,f=1.6*fr
TdRdLdCd (c) (d)
Pressure (Pa) -28 -26 -24 -22 -20 D i ss i p a t e d po w e r ( W ) R0=1 m coated bubble,f=2*fr
TdRdLdCd (e) (f)
Pressure (Pa) -28 -26 -24 -22 -20 -18 D i ss i p a t e d po w e r ( W ) R0=1 m,f=3*fr
TdRdLdCd (g) (h)
Fig. 4. Bifurcation structure (left column) and the dissipated power as a function ofpressure (right column) of the oscillations of a coated C3F8 bubble with R = 1 µm for f = 1 . f r (a-b)- f = 1 . f r (c-d)- f = 2 f r (e-f) & f = 3 f r (g-h). a b s (( R . (t)) m / s R0=10 m uncoated bubble (a) (b) R d ( W ) -14 R0=10 m uncoated bubble f=0.25*frf=0.5*frf=0.9*frf=1*frf=1.2*frf=1.75*frf=2*frf=3*fr t o t a l d i ss i p a t e d po w e r ( W ) -14 R0=10 m uncoated bubble f=0.25*frf=0.5*frf=0.9*frf=1*frf=1.2*frf=1.75*frf=2*frf=3*fr (c) (d) S T DR R0=10 m uncoated bubble f=0.25*frf=0.5*frf=0.9*frf=1*frf=1.2*frf=1.75*frf=2*frf=3*fr (e)
Fig. 5. Nondestructive (
R/R ≤
2) values of: a) | ˙ R ( t ) | max ( V m ), b) Maximumbackscattered pressure ( | P sc | max ( P m ), c) Rd, d) W total and e) STDR as a functionof pressure in the oscillations of an uncoated air bubble with R = 10 µm . | ˙ R ( t ) | max , | P sc | max , total dissipated power and STDR In the previous subsection we investigated the evolution of the Cd, Ld, Rd andTd as a function of pressure at different frequencies and related their changes tothe nonlinear behavior of the bubble. In this section we consider only stable non-destructive bubble oscillations ( R max R ≤ | ˙ R ( t ) | max ), maximum amplitude of the re-radiated20a) (b) R d ( W ) -18 R0=1 m coated bubble f=0.25*frf=0.45*frf=0.8*frf=1*frf=1.2*frf=1.6*frf=2*frf=3*fr t o t a l d i ss i p a t e d po w e r ( W ) -18 R0=1 m coated bubble f=0.25*frf=0.45*frf=0.8*frf=1*frf=1.2*frf=1.6*frf=2*frf=3*fr (c) (d) S T DR R0=1 m coated bubble f=0.25*frf=0.45*frf=0.8*frf=1*frf=1.2*frf=1.6*frf=2*frf=3*fr (e)
Fig. 6. Nondestructive (
R/R ≤
2) values of: a) | ˙ R ( t ) | max ( V m ), b) Maximumbackscattered pressure ( | P sc | max ( P m ), c) Rd, d) W total and e) STDR as a functionof pressure in the oscillations of a coated C3F8 bubble with R = 2 µm . pressure ( | P sc | max ), total dissipated power ( W total = Rd + Ld + T d + Cd ), andthe scattering to dissipation ratio (STDR) when bubble is sonicated with thefrequencies that are studied in Figs. 1-4 and for excitation pressures belowthe bubble destruction threshold. | ˙ R ( t ) | max is abbreviated with V m from hereon for simplicity and | P sc | max is abbreviated with P m . ST DR = RdRd + Ld + T d + Cd where Cd is equal to zero for the uncoated bubble.Fig. 5a plots V m as a function of the excitation pressure for different frequen-cies for an uncoated air bubble with R = 10 µm . V m is only presented forthe oscillation regimes that most probably results in non-destructive bubble21scillations R/R ≤ f = 0 . f r , V m reaches the maximum value for non-destructive oscillations. Thiscan have advantages in drug delivery applications since higher wall velocityresults in faster streaming and lower frequency of oscillations leads to smallervalues for the thickness of the boundary layer [63]. Since shear stress on thenearby objects is proportional to wall velocity and inversely proportional tothe thickens of the boundary layer [64] sonication in this regime may result inhigher shear stress values compared to other frequencies.Fig. 5b shows P m as a function of pressure for the studied frequencies inFigs. 1-2. Sonication with f = 3 f r leads to the highest P m ; thus if the goalof the application is to increase the absolute amplitude of P sc and enhanceechogenecity then sonication with f = 3 f r and pressures above the pressurethreshold for generation of P3 oscillations will be the optimized frequency andpressure.Rd can be used as a measure for continuous bubble activity. In contrast to P m which denotes the maximum spontaneous back-scattered (re-radiated) pressure,Rd is a measure of sustained bubble activity as it is averaged over time. Fig.5c shows that the maximum non-destructive Rd occurs for f = 0 . f r andfrequencies below resonance display the largest non-destructive Rd.The total dissipated power ( W total = Rd + Ld + T d ) is shown in Fig. 5d.Maximum dissipated power occurs for f = 0 . f r . Sonication below resonancehave the advantage of higher dissipated powers. This is useful in applicationswhere bubbles are used to enhance the power deposition by ultrasound toincrease the generated heating. Furthermore, when compared to f = f r , below100 kPa, the dissipated power is lower when f = 0 . f r and above 100 kPa thedissipated power undergoes a sharp increase and becomes approximately 4.8times larger than the case of sonication with f = f r . This has advantages infocused ultrasound heating enhancement; by taking advantage of the sharppressure gradients of the focused ultrasound transducers, spatial heating priorto the focal point is limited and heating at the focal region can be enhanced.Fig. 5e displays the STDR as a function of pressure. Frequencies above f r have larger STDR; the higher the freqeuncy the larger is the STDR whichleads to the maximum STDR at f = 3 f r . It is interesting to note that forfrequencies above resonance the onset of non-linear oscillations results in adecrease in STDR; STDR then grows as pressure increases. This is mainlydue to the increase in other damping parameters especially Td. As pressureincreases, due to the faster growth rate of Rd, STDR raises again. Freqeunciesbelow resonance have lower STDRs compared to f = f r . This is because ofthe increased Td and decreased Rd; however, the onset of SN bifurcation isconcomitant with an increase in STDR for f = 0 . f r (PDfr) and f = 0 . f r (2nd SuH) oscillations.The case of the C3F8 coated bubble with R = 1 µm is shown in Fig. 6. Sameconclusions can be drawn as the case of the uncoated bubble in Figs. 5. Thispossibly indicates a universal behavior in the studied parameters for the casesconsidered in this paper. 22 Summarizing points
Acoustic waves are highly dissipated when they pass through bubbly media.Dissipation by bubbles takes place through thermal damping (Td), radiationdamping (Rd) and damping due to the friction of the liquid (Ld) and frictionof the coating (Cd). Td, Rd, Ld and Cd are nonlinear and depend on thecomplex dynamics of the bubbles. The correct estimation of dissipation eventsin the bubble oscillations will help in optimizing the relevant applications bymaximizing a desirable parameter.Most previous studies were limited by linear approximations [37, 66, 67, 68, 69,70, 71]. These approximations lead to inaccurate estimation of the dissipationphenomenon in applications as they are only valid for low pressures and linearregimes of low amplitude oscillations. Despite the importance of understandingthe nonlinear dissipation events; only a few recent studies have attempted toinvestigate the problem accounting for the full non-linearity of the subject[38, 39, 40, 41, 46, 47].At present, the pressure dependence of the dissipation events is not well un-derstood. Thus, in this paper we attempted to classify the bubble oscillationsat various excitation frequencies as a function of pressure. Using a recentcomprehensive approach [50, 56] in studying the nonlinear bubble dynamics wehave classified the nonlinear oscillations of the uncoated and coated bubblesas a function of pressure excited with frequencies that result in 3rd and 2ndSuH regimes, pressure dependent (PD) resonance ( f r ) oscillations, bubblessonicate with f r (linear resonance frequency), PD subharmonic (SH) resonance( P Df sh ), 1 / f sh ) and 1 / f = 3 f r ).We have considered the nonlinear thermal and radiation effects in modeling theoscillations of the bubbles. Dynamics of the bubble including the generation of(2nd and 3rd order) SuH and (7 / /
1- When a bubble (coated or uncoated) is sonicated with a frequency which isapproximately between f r and f r , and above a pressure threshold 3 maxima23 issipation mechanisms when f = 0 . − . f r Oscillation shape Linear P1(3 max-ima) P2(6 max-ima) chaos giant 2ndSuH reso-nanceUncoated air bub-ble with R =10 µm T d > Ld >Rd T d > Rd = Ld T d = Rd >Ld Rd > T d >Ld Rd > Ld >T d
Uncoated air bub-ble with R =2 µm T d (cid:117) Ld >Rd T d (cid:117) Ld (cid:117) Rd Ld (cid:117)
Rd >T d Rd > Ld >T d Rd > Ld >T d coated C3F8 bub-ble with R =4 µm Cd > Ld (cid:117) T d > Rd Cd > Ld (cid:117)
Rd > T d Cd > Rd (cid:117)
Ld > T d Cd (cid:117)
Rd >Ld > T d Rd > Cd >Ld > T d coated C3F8 bub-bles with R =4 µm Cd > Ld >T d > Rd Cd > Ld >Rd > T d Cd > Rd (cid:117) Ld > T d Cd (cid:117)
Rd >Ld > T d Rd > Cd >Ld > T d
Table 2Evolution of dissipation powers at different nonlinear regimes for uncoated air andcoated C3F8 bubbles when f = 0 . − . f r occur in the P1 oscillations. In this regime, the third harmonic componentof the scattred pressure ( P sc ) is larger than the other frequency components.This is suitable for applications like SuH imaging [72, 73]. Above a secondpressure threshold period doubling (Pd) takes place and oscillations becomeP2 with 6 maxima. At this point the 7 / P sc is largerthan the SH and other UHs components of the signal. This frequency andpressure range is suitable for high resolution UH imaging [74, 75, 76] and passivecavitation mapping techniques [77]. Further pressure increases beyond thisregion most probably results in bubble destruction as R/R exceeds 2. Pressureincrease also result in the generation of chaos and periodic oscillations of higheramplitudes. At higher pressures a giant resonance (2nd order SuH) emerges outof the chaotic window. The corresponding evolution of the dissipative powersat different nonlinear regimes are summarized in table 2.2- When f (cid:117) . − . f r both of the cases of the coated and uncoated bubblesstart with P1 oscillations and above a pressure threshold a second maximaoccur in the P1 oscillations. This is simultaneous with a fast growth of the2nd harmonic component of the P sc (2nd harmonic component becomes thestrongest frequency component in the spectra). This frequency and pressurerange is suitable for 2nd harmonic imaging applications of ultrasound [72, 73].Above a second pressure threshold Pd happens and oscillations become P2 with4 maxima. At this point between the SH and UH components of the frequencyspectrum of the P sc , the 5 / issipation mechanisms when f = 0 . − . f r Oscillation shape Linear P1(2 max-ima) P2(4 max-ima) chaos P1 giant res-onanceUncoated air bub-ble with R =10 µm T d > Ld >Rd T d > Rd (cid:117) Ld T d (cid:117)
Rd >Ld Rd > T d >Ld Rd > Ld >T d
Uncoated air bub-ble with R =2 µm T d (cid:117) Ld >Rd T d (cid:117)
Ld >Rd Ld (cid:117)
Rd >T d Rd > Ld >T d Rd > Ld >T d coated C3F8 bub-ble with R =4 µm Cd > Ld (cid:117) T d > Rd Cd > Ld (cid:117)
Rd > T d Cd > Rd (cid:117)
Ld > T d Cd > Rd >Ld > T d Rd (cid:117)
Cd >Ld > T d coated C3F8 bub-ble with R =1 µm Cd > Ld >T d > Rd Cd > Ld >Rd > T d Cd > Rd (cid:117) Ld > T d Cd > Rd >Ld > T d Rd (cid:117)
Cd >Ld > T d
Table 3Evolution of dissipation powers at different nonlinear regimes for uncoated air andcoated C3F8 bubbles when f = 0 . − . f r . ultrasound [74, 75, 76] and passive cavitation mapping techniques [77] or highresolution treatment monitoring using UH emissions [78]. Further pressureincrease results in chaos and R/R exceeding 2. Pressure increase beyond thispoint leads to emergence of a P1 giant resonance out of the chaotic window.The corresponding evolution of the dissipative powers at different nonlinearregimes are summarized in table 3.3- When f = 0 . − . f r (the pressure dependent resonance frequency ( P Df r [32])), at lower pressures oscillations are P1 with 1 maximum. Above a pressurethreshold, a saddle node (SN) bifurcation takes place and P1 oscillationsundergo a large increase in amplitude to another P1 oscillation with 1 maximum.At SN, the maxima curve and the Poincar curve has the same value (the wallvelocity is in phase with the excitation force). The Poincar and maximacurves diverge from each other as pressure increases beyond the SN. Theoccurrence of the SN can have significant advantages in imaging techniquesbased on amplitude modulation [32, 74, 79, 80]. Beyond the SN, the increase inexcitation pressure leads to a monotonic increase in oscillation amplitude andabove a second pressure threshold Pd takes place and oscillations become P2with 2 maxima. Apart from the coated bubble with R = 1 µm , other studiedbubbles in this frequency range most probably are destroyed as R max R exceeds 2before any P2 is generated. Further pressure increase leads to successive Pds tochaos with a possible window (of P3/P6 oscillations with 3/6 maxima) whichis located inside chaotic window. At higher pressures a P2 giant resonancemay emerge out of the chaotic window. The corresponding evolution of thedissipative powers at different nonlinear regimes are summarized in table 4.25 issipation mechanisms when f = 0 . − . f r Oscillation shape Linear SN-P1(1maxima) P2(2 max-ima) chaos P2 giant res-onanceUncoated air bub-ble with R =10 µm T d > Ld >Rd T d > Rd (cid:117) Ld T d (cid:117)
Rd >Ld Rd > T d >Ld Rd > Ld >T d
Uncoated air bub-ble with R =2 µm T d (cid:117) Ld >Rd T d (cid:117)
Ld >Rd Ld (cid:117)
Rd >T d Rd > Ld >T d Rd > Ld >T d coated C3F8 bub-ble with R =4 µm Cd > Ld >Rd > T d Cd > Ld (cid:117) Rd > T d Cd > Rd >Ld > T d Cd > Rd >Ld > T d Rd (cid:117)
Cd >Ld > T d coated C3F8 bub-ble with R =1 µm Cd > Ld >Rd > T d Cd > Ld >Rd > T d Cd > Ld >Rd > T d Cd > Rd >Ld > T d Rd (cid:117) Cd >Ld > T d
Table 4Evolution of dissipation powers at different nonlinear regimes for uncoated air andcoated C3F8 bubbles when f = 0 . − . f r .
4- When f = f r (linear resonance frequency of the bubbles) oscillations areP1 with 1 maxima at lower pressures. At the beginning of the bifurcationdiagrams; the curve that is constructed with the method of peaks has theexact value as the Poincar method. This indicates that the wall velocity isin phase with the driving signal. As the pressure increases the two curvesstart diverging as the resonance frequency changes with pressure. At higherpressures resonance frequency shifts to smaller values [32] ( P df r ). Above apressure threshold, bubbles undergo Pd and oscillations become P2 with twomaxima. For coated bubbles, Pd can occur when R max R < R max R > f r ). The P2 oscillations hasbeen extensively studied in our previous work [58] without the inclusion ofthermal damping effects. P2 oscillations undergo successive Pd to chaos. Athigher pressures a P2 (with two maxima) giant resonance may emerge out ofthe chaotic window. The giant resonance oscillations undergo successive Pdsto chaotic oscillations of higher amplitude. The corresponding evolution of Cd,Rd, Ld and Td are summarized in table 5.5- When f = 1 . f r , at lower excitation pressures, oscillations are P1 with 1maximum. Contrary to the case of f = f r , the Poincar curve and the curveconstructed by method of peaks start diverging right at the beginning of thebifurcation diagram. Increasing pressure leads to Pd and oscillations becomesP2 with two maxima. At this frequency P2 oscillations can be non-destructive(in case of both coated and uncoated bubbles) as when Pd occurs since R max R issipation mechanisms when f = f r Oscillation shape Linear reso-nance Linear P2(2 max-ima) chaos P2 giant res-onanceUncoated air bub-ble with R =10 µm T d > Ld >Rd T d > Rd (cid:117) Ld T d (cid:117)
Rd >Ld Rd > T d >Ld Rd > Ld >T d
Uncoated air bub-ble with R =2 µm T d > Ld >Rd Ld > Rd Ld (cid:117) Rd >T d Rd > Ld >T d Rd > Ld >T d coated C3F8 bub-ble with R =4 µm Cd > Ld >Rd > T d Cd > Ld (cid:117) Rd > T d Cd > Rd >Ld > T d Cd > Rd >Ld > T d Rd (cid:117)
Cd >Ld > T d coated C3F8 bub-ble with R =1 µm Cd > Ld >Rd > T d Cd > Ld >Rd > T d Cd > Ld >Rd > T d Cd > Rd >Ld > T d Rd (cid:117) Cd >Ld > T d
Table 5Evolution of dissipation powers at different nonlinear regimes for uncoated air andcoated C3F8 bubbles when f = f r . is below 2. Sonication with this frequency and in pressure ranges responsiblefor P2 oscillations may lead to 3 / f = 1 . − . f r (pressure dependent subharmonic (SH) resonancefrequency P Df sh [58]) at lower pressures oscillations are P1 with 1 maxima.Pressure increase leads to one of the following scenarios: 1- generation of Pdabove a pressure threshold; then, a SN bifurcation from P2 oscillations oflower amplitude to P2 oscillations (1 maximum) of higher amplitude. 2- SNbifurcation above a pressure threshold from P1 oscillations with 1 maxima toP2 oscillations with 1 maxima. This happens while R max R is below 2; therefore,the bubbles may sustain non-destructive P2 oscillations when insonated withfrequencies between f = 1 . − . f r . Further pressure increases lead to thegeneration of a second maximum (with the same amplitude of the higherbranch of the Poincar curve). Occurrence of the SN can provide significantadvantages for amplitude modulation techniques [78, 79, 80] and in this casebecause of the higher sonication frequency, we can expect higher resolution.Moreover, we have shown in [59] that occurrence of SN leads to oversaturation27 issipation mechanisms when f = 1 . f r Oscillation shape Linear Linear P2(2 max-ima) chaos chaos/P2giant reso-nanceUncoated air bub-ble with R =10 µm T d > Ld (cid:117) Rd T d > Rd >Ld T d > Rd >Ld Rd (cid:117)
T d >Ld Rd > Ld >T d
Uncoated air bub-ble with R =2 µm T d > Ld >Rd Ld > T D >Rd Ld > T d >Rd Ld (cid:117) Rd >T d Rd > Ld >T d coated C3F8 bub-ble with R =4 µm Cd > Ld >Rd > T d Cd > Ld (cid:117) Rd > T d Cd > Rd >Ld > T d Cd > Rd >Ld > T d Cd (cid:117)
Rd >Ld > T d coated C3F8 bub-ble with R =1 µm Cd > Ld >Rd > T d Cd > Ld >Rd > T d Cd > Ld >Rd > T d Cd > Rd >Ld > T d Rd (cid:117) Cd >Ld > T d
Table 6Evolution of dissipation powers at different nonlinear regimes for uncoated air andcoated C3F8 bubbles when f = 1 . f r . of the 1 / / P sc . This can providehigher contrast to tissue and signal to noise ratio in SH imaging techniques[81, 82, 83] and possibly SH and UH monitoring of treatments [77, 78, 84, 85].Oscillations undergo successive Pds to chaos. Further pressure increase maylead to the emergence of a P3 giant resonance which will undergo successivePds to chaotic oscillations of higher amplitude. The dynamics of the bubblesonicated with their P Df sh (in the absence of thermal damping) has beenextensively studied in our previous work [59]. The corresponding evolution ofdissipative powers are summarized in table 7.7- when f = 2 f r (linear SH resonance frequency f sh [58]) oscillations are P1with 1 maximum at lower pressures. Above a pressure threshold Pd takes placeand oscillations become P2 with 2 maxima. As pressure increases one of themaxima disappears with pressure increase and the P2 oscillations evolve inthe form of a bowtie (the curve that is constructed using the Poincar method).Later the second maxima re-appears with an amplitude equal to the higherbranch of the Poincar curve. Oscillations undergo successive Pds to chaos. When f = 2 f r , P2 oscillations occur for the widest excitation pressure range and R max R is below 1.5; therefore, bubbles have the highest probability of sustainingnon-destructive P2 oscillations. Analytical solutions [86, 87, 88, 89, 90, 91]predict the generation of P2 oscillations at the lowest pressure threshold whenthe bubble is sonicated with f = 2 f r . Later, it was shown in [92, 93] in case ofsmaller bubbles (e.g. R = 0 . µm ) that the lowest pressure threshold occurswhen bubble is sonicated with a frequency near its f r . They concluded that28 issipation mechanisms when f = 1 . − . f r Oscillation shape Linear P2 throughSN P4(4 maxima) chaos chaos/P3 giantresonanceUncoated air bub-ble with R =10 µm T d > Rd >Ld Rd >T D > Ld Rd > T d > Ld Rd > T d >Ld Rd > Ld >T d ( Gf r )Uncoated air bub-ble with R =2 µm T d (cid:117) Ld >Rd Ld > Rd >T d Ld (cid:117) Rd ¿Td Ld (cid:117) Rd >T d Rd > Ld > T d coated C3F8 bub-ble with R =4 µm Cd > Rd >Ld > T d Cd > Rd >T d Cd > Ld (cid:117) Rd > Ld > T d Cd > Rd >Ld > T d Cd (cid:117)
Rd >Ld > T d coated C3F8 bub-ble with R =1 µm Cd > Ld (cid:117) Rd > T d Cd > Ld (cid:117)
Rd > T d Cd > Ld (cid:117)
Rd > T d Cd > Ld (cid:117)
Rd > T d Cd > Rd >Ld > T d
Table 7Evolution of dissipation powers at different nonlinear regimes for uncoated air andcoated C3F8 bubbles when f = 1 . − . f r . the increased damping is responsible for shift the lowest frequency threshold.However, none of the previous studies included both of the pressure dependentthermal and radiation damping effects. In this work, we have included both ofthese effects with their full non-linearity and observed that the lowest pressurethreshold of P2 oscillations occurs at none of the f = f r or f = 2 f r , but itoccurs at frequencies below P DF f r . As an instance, for the uncoated air bubble R = 10 µm pressure thresholds for P2 oscillations are 87.5, 82, 88, 170 and96kPa respectively at f=0.25, 0.35, 0.5, 1 and 2 f r . Thus for this bubble lowestP2 pressure threshold is 0 . f r . This may be explained with the increaseddamping effects due to thermal damping and pressure dependent non-linearcoupling. The study of the lowest pressure threshold for P2 oscillations andthe reasons behind it is not within the scope of this paper and can be thesubject of future studies. The corresponding evolution of dissipation powersare summarized in table 8.8- When f = 3 f r oscillations are P1 with 1 maximum at lower pressures.Oscillation amplitude grow very slowly withe excitation pressure increase andabove a pressure threshold P3 oscillations of higher amplitude are generatedthrough a SN bifurcation. Later P3 oscillations undergo Pd to P12 followedby successive Pds to a small chaotic window before the oscillations convertto P1 with lower amplitude. The corresponding evolution of the dissipativepowers is summarized in table 9. The SN bifurcation is concomitant with asharp increase in P sc . This has advantages for amplitude modulation imagingtechniques [74, 79, 80] at higher frequencies [94]. The pressure amplitudes forthe pulses shoudl be chosen below and above the SN pressure.29 issipation mechanisms when f = 2 f r Oscillation shape Linear P2 throughSN P4(4 maxima) chaos chaos/P3 giantresonanceUncoated air bub-ble with R =10 µm T d > Rd >Ld T D >Rd > Ld T d (cid:117) Rd > Ld T d (cid:117)
Rd >Ld Rd > Ld >T d ( Gf r )Uncoated air bub-ble with R =2 µm T d (cid:117) Ld >Rd T d (cid:117)
Ld >Rd Ld (cid:117) Rd ¿Td Ld (cid:117) Rd >T d Rd > Ld > T d coated C3F8 bub-ble with R =4 µm Cd > Rd >Ld > T d Cd > Rd >Ld > T d Cd > Rd >Ld > T d Cd (cid:117) Rd >Ld > T d Rd > Cd >Ld > T d coated C3F8 bub-ble with R =1 µm Cd (cid:117) Rd >Ld > T d Cd > Ld (cid:117)
Rd > T d Cd > Rd >Ld > T d Cd (cid:117)
Rd >Ld > T d Cd (cid:117)
Rd >Ld > T d
Table 8Evolution of dissipation powers at different nonlinear regimes for uncoated air andcoated C3F8 bubbles when f = 2 f r .Dissipation mechanisms when f = 3 f r Oscillation shape Linear P3 throughSN P6(6 maxima) chaos linearUncoated air bub-ble with R =10 µm Rd > T d >Ld Rd > T d >Ld Rd > T d > Ld Rd > T d >Ld Rd > T d >Ld Uncoated air bub-ble with R =2 µm Rd > Ld >T d Rd (cid:117) Ld >T d Rd > Ld > T d Rd > Ld >T d Rd > Ld >T d coated C3F8 bub-ble with R =4 µm Rd > Cd >Ld > T d Cd > Rd >Ld > T d Cd > Rd > Ld >T d Cd (cid:117) Rd >Ld > T d Rd (cid:117)
Cd >Ld > T d coated C3F8 bub-ble with R =1 µm Rd > Cd >Ld > T d Rd > Cd >Ld > T d Rd > Cd > Ld >T d Rd > Cd >Ld > T d Rd > Cd >Ld > T d Table 9Evolution of dissipation powers at different nonlinear regimes for uncoated air andcoated C3F8 bubbles when f = 3 f r .
9- within the pressure ranges that were investigated here, occurrence of thegiant resonances were in the form of a large amplitude periodic oscillationsthat emerge out of the chaotic window at higher pressures. These oscillationswere concomitant with a sharp increase in Rd, Ld and Cd (in case of coated30ubbles) and at the same time concomitant with a decrease in Td. Thisimplies that oscillations have larger wall velocity amplitudes and acceleration;moreover due to the larger instantaneous changes of the R max to R min highercore temperatures are expected. The faster collapses and rebound in theseoscillation regimes leaves very little time for heat transfer thus Td decreases.This may advantages for sonochemical applications of ultrasound as highercore temperatures are achieved and thermal loss is decreased. | ˙ R ( t ) | max , | P sc | max , Rd, W total and STDR during non-destructive oscilla-tions and their possible applications In this section we summarize some important parameters related to the bubblebehavior and link them to possible medical and sonochemical applications.These parameters are extracted for excitation pressures that leads to non-destructive regime of oscillations ( R max R ≤
2) and for exciation freqeuncy rangeof f = 0 . f r − f r .1- The maximum wall velocity amplitude ( | ˙ R ( t ) | max ) was the largest for thebubbles that were sonicated with f = 0 . − . f r . Higher wall velocities re-sults in faster micro-streaming. The shear stress induced by bubbles on nearbyobjects is proportional to the micro-streaming velocity and to the thickness ofthe boundary layer [63, 64, 96]. The thickness of the boundary layer is inverselyproportional to frequency [63, 64, 96]. Thus, sonication with f = 0 . − . f r not only can produce the highest non-destructive micro-streaming, but it alsohas a small boundary layer. Sonication in this frequency range may thereforeenhance the shear stress and drug delivery efficacy. Moreover, non-destructiveand non-inertial high shear stresses in this frequency may enhance the surfacecleaning [97, 98, 99] while avoiding damage to delicate micro-structures (e.g.semi-conductor industry, optical devices & precision apparatus) usually stem-ming from violent inertial collapse of bubbles [98, 99]. Quantification of theshear stress at various non-linear regimes is a complex task and is the subjectof future studies.2- The maximum amplitude of the back-scattred pressure ( | P sc | max ) frombubbles was the largest for bubbles sonicated with 3 f r . Echogencity of theultrasound images is directly proportional to | P sc | max . Thus, in applications likeB-Mode imaging [82] using contrast agents, higher frequencies ( f = 1 . − f r )may be desired. However, one must also note that the higher | P sc | max occurs ata higher pressure for f = 3 f r ; thus, the signal intensity from the backgroundtissue can be higher. In the absence of any non-linear signal acquisition (asan instance amplitude modulation [74, 79, 80] or phase inversion [100]) thatsuppresses the tissue response in the final image, the effect of higher scatteringfrom tissue at higher pressures should also be considered. On the other hand theabrupt increase in the | P sc | max of the bubble when SN bifurcation takes place(e.g. at f = 1 . f r or f = 3 f r (Fig.12b)) can be used to increase the residual31ignal in amplitude modulation techniques and increase the contrast to tissueratio and signal to noise ratio. In amplitude modulation technique two pulsesare sent to the target with different amplitudes. The received signals from thetarget are then scaled and subtracted; due to the linear tissue response the twosignals cancel each other after subtraction. However, the nonlinear responseof the bubbles with respect to increase in pressure results in a considerableresidue which leads to enhanced CTR. Another application for the non-inertialhigher P sc can be in drug delivery or surface cleaning. The increased pressureradiated by the bubbles can increase the permeability [96] of the cells or objectsin their vicinity and contribute to the drug delivery enhancement or cleaning.3- Rd and W total were maximum for bubbles that were sonicated with 0 . − . f r . Higher Rd and W total are of great importance for applications relatedto bubble enhanced heating in high intensity focused ultrasound (HIFU)[30, 101, 102] and ultrasound thermal therapies and hyperthermia. Enhancedheating is of particular interest especially in cases like liver and brain wherethere is strong cooling of tissue due to high blood perfusion and the presenceof skull [103] and rib cage limits the amount of ultrasound energy that canbe delivered to target. In [104], it was shown that enhancing the depositedpower by increasing the wave dissipation or enhancing the pressure amplitudecan decrease the effect of blood flow cooling until full necrosis takes place.Sonication with f = 0 . − . f r can provide Rd and W total of at least 6times greater when compared to the case of sonication with f r . Moreover, thehigher frequency component of the P sc signal (e.g. 3rd SuH, 7 / f = 0 . − . f r is that Rd and W total are very small for low pressures; however, above a pressure threshold(concomitant with the generation of UHs and SHs in the P sc ) Rd and W total significantly increase. This finding is in line with experimental observation[30], where enhanced heating was concomitant with SH and UH emissions andbroadband noise. The lower dissipation of acoustic waves below the pressurethreshold leads to minimum enhanced heating and wave dissipation in thepre-focal tissue [32, 105, 106, 107, 108]. This allows higher energy delivery forbubbles in the target (especially in cases where delivery of higher acousticenergy is challenging) and enhances the safety of the treatment as the off-target bubble activity is minimized. Moreover, the generation of UHs at thetarget can be used to monitor and control the treatment using methods likepassive cavitation detection [77, 85]. W total and Rd were minimum for higherfrequencies f = 1 . − f r . Thus, in addition to higher P sc which leads to higherechogenecity in ultrasonic imaging, sonication with these frequencies resultsin lower heating due to bubble activity. This is another reason why higherfrequencies may be more suitable for contrast enhanced ultrasound imaging.Moreover, enhanced absorption ( W total ) in the target can be used to shield [34]structures with higher ultrasound attenuation (as an instance post-target bone[104] in brain).4- The STDR as a function of pressure is nonlinear. The highest STDR belongs32o f = 3 f r . In the absence of super-harmonic resonance, generation of SHsand UHs are concomitant with a decrease in STDR. As it was discussed inprevious sections for f = 1 . − f r , Td undergoes a large increase when SHsare generated which consequently leads to a decrease in STDR. Despite thedecrease, STDR still remains higher than f < f r . The higher STDR havegreat advantages for contrast enhanced imaging. Higher STDR means bubblescatters more and dissipates less. This has consequences in increasing theechogenecity of the target and the underlying tissue. However, higher STDRby itself does not imply that a set of exposure parameters are suitable forimaging applications. As an instance when f = 3 f r STDR is very high atlower pressures (e.g. 10 kPa); however, at the same time P sc is very small. Thismeans that despite a high STDR, because of the weaker scattering by thebubble, the contrast signal may not be distinguishable from the backgroundnoise. Thus, STDR should be used in tandem with the P sc and Rd curves tostudy the suitable exposure parameters for the relevant application. In this work we investigated the mechanisms of energy dissipation in bubbleoscillations and their contribution to the total damping ( W total ) at variousnonlinear regimes of bubble oscillations. By using a comprehensive bifurcationanalysis, we have classified the nonlinear dynamics of the bubbles and thecorresponding dissipation mechanisms. The bifurcation structure of the un-coated and coated bubbles including the full thermal and radiation effects havebeen classified for the first time. Using our recently developed equations forenergy dissipation in the oscillations of coated and uncoated bubbles[40, 41],the pressure dissipation mechanisms of ultrasonic energy were analyzed indetail. Results were presented in tandem with the bifurcation diagrams andseveral nonlinear features of dissipation phenomenon were revealed and clas-sified. We have shown that by choosing suitable frequency and pressure aparticular bubble related effect can be enhanced (e.g. maximum wall veloc-ity amplitude, maximum scattered pressure, etc.). The exposure parametersby which each of these parameters are maximum seem to be universal andregardless of the bubble size and coating. For example within the exposureparameter ranges that were studied in this paper we show that, for all thebubbles maximum non-destructive wall velocity occurs when the bubble issonicated with 0 . f r ≤ f ≤ . f r , maximum non-destructive scattered pressureand STDR are reached when f = 3 f r , nondestructive radiation damping andtotal scattered power are maximized at f = 0 . rf r . These parameters canbe used as a guideline to optimize possible related applications (e.g. Imaging,drug delivery, surface cleaning, etc.). 33 Acknowledgments
The work is supported by the Natural Sciences and Engineering Research Coun-cil of Canada (Discovery Grant RGPIN-2017-06496), NSERC and the CanadianInstitutes of Health Research ( Collaborative Health Research Projects ) andthe Terry Fox New Frontiers Program Project Grant in Ultrasound and MRIfor Cancer Therapy (project
References [1] Lauterborn, Werner, and Thomas Kurz. ”Physics of bubble oscillations.” Reportson progress in physics 73, no. 10 (2010): 106501.[2] Parlitz, U., V. Englisch, C. Scheffczyk, and W. Lauterborn. ”Bifurcation structureof bubble oscillators.” The Journal of the Acoustical Society of America 88, no.2 (1990): 1061-1077.[3] Lauterborn, Werner, and Joachim Holzfuss. ”Acoustic chaos.” InternationalJournal of bifurcation and Chaos 1, no. 01 (1991): 13-26.[4] Holzfuss, Joachim, Matthias Rggeberg, and Andreas Billo. ”Shock wave emissionsof a sonoluminescing bubble.” Physical review letters 81, no. 24 (1998): 5434.[5] Holzfuss, Joachim. ”Acoustic energy radiated by nonlinear spherical oscillationsof strongly driven bubbles.” Proceedings of the Royal Society A: Mathematical,Physical and Engineering Sciences 466, no. 2118 (2010): 1829-1847.[6] Dollet, Benjamin, Philippe Marmottant, and Valeria Garbin. ”Bubble dynamicsin soft and biological matter.” Annual Review of Fluid Mechanics 51 (2019):331-355.[7] D’Onofrio, Mirko, Stefano Crosara, Riccardo De Robertis, Stefano Canestrini,and Roberto Pozzi Mucelli. ”Contrast-enhanced ultrasound of focal liver lesions.”American journal of roentgenology 205, no. 1 (2015): W56-W66.[8] K. Ferrara, R. Pollard, and M. Borden, Ultrasound microbubble contrast agents:fundamentals and application to gene and drug delivery,
Annu. Rev. Biomed.Eng.
InternationalJournal of Hyperthermia
11] M.A. OReilly and K. Hynynen, Blood-brain barrier: real-time feedback-controlledfocused ultrasound disruption by using an acoustic emissions-based controller,
Radiology
TheJournal of the Acoustical Society of America , 132(1) (2012) 544-553.[13] Soluian, Stepan Ivanovich. Theoretical foundations of nonlinear acoustics.Consultants Bureau, 1977.[14] Rivas, David Fernandez, Bram Verhaagen, James RT Seddon, Aaldert G. Zijlstra,Lei-Meng Jiang, Luc WM van der Sluis, Michel Versluis, Detlef Lohse, and HanJGE Gardeniers. ”Localized removal of layers of metal, polymer, or biomaterialby ultrasound cavitation bubbles.” Biomicrofluidics 6, no. 3 (2012).[15] Maisonhaute, Emmanuel, Cesar Prado, Paul C. White, and Richard G. Compton.”Surface acoustic cavitation understood via nanosecond electrochemistry. PartIII: Shear stress in ultrasonic cleaning.” Ultrasonics sonochemistry 9, no. 6(2002): 297-303..[16] Ohl, Claus-Dieter, Manish Arora, Rory Dijkink, Vaibhav Janve, and Detlef Lohse.”Surface cleaning from laser-induced cavitation bubbles.” Applied physics letters89, no. 7 (2006): 074102.[17] Roovers, Silke, Tim Segers, Guillaume Lajoinie, Joke Deprez, Michel Versluis,Stefaan C. De Smedt, and Ine Lentacker. ”The role of ultrasound-drivenmicrobubble dynamics in drug delivery: from microbubble fundamentals toclinical translation.” Langmuir (2019).[18] Kooiman, Klazina, Hendrik J. Vos, Michel Versluis, and Nico de Jong. ”Acousticbehavior of microbubbles and implications for drug delivery.” Advanced drugdelivery reviews 72 (2014): 28-48.[19] Marmottant, Philippe, and Sascha Hilgenfeldt. ”Controlled vesicle deformationand lysis by single oscillating bubbles.” Nature 423, no. 6936 (2003): 153.[20] M.A. OReilly and K. Hynynen, Blood-brain barrier: real-time feedback-controlledfocused ultrasound disruption by using an acoustic emissions-based controller,
Radiology
23] Johnsen, Eric, and Tim Colonius. ”Shock-induced collapse of a gas bubble inshockwave lithotripsy.” The Journal of the Acoustical Society of America 124,no. 4 (2008): 2011-2020.[24] Gong, Cuiling, and Douglas P. Hart. ”Ultrasound induced cavitation andsonochemical yields.” The Journal of the Acoustical Society of America 104, no.5 (1998): 2675-2682.[25] Suslick, Kenneth S. ”Sonochemistry.” science 247, no. 4949 (1990): 1439-1445.[26] Suslick, Kenneth S., S. J. Doktycz, and E. B. Flint. ”On the origin ofsonoluminescence and sonochemistry.” Ultrasonics 28, no. 5 (1990): 280-290.[27] Mason, Timothy J., Larysa Paniwnyk, and J. P. Lorimer. ”The uses of ultrasoundin food technology.” Ultrasonics sonochemistry 3, no. 3 (1996): S253-S260.[28] Canavese, Giancarlo, Andrea Ancona, Luisa Racca, Marta Canta, BiancaDumontel, Federica Barbaresco, Tania Limongi, and Valentina Cauda.”Nanoparticle-assisted ultrasound: a special focus on sonodynamic therapyagainst cancer.” Chemical Engineering Journal 340 (2018): 155-172.[29] Holt, R. Glynn, D. Felipe Gaitan, Anthony A. Atchley, and Joachim Holzfuss.”Chaotic sonoluminescence.” Physical review letters 72, no. 9 (1994): 1376.[30] Holt, R. Glynn, and Ronald A. Roy. ”Measurements of bubble-enhancedheating from focused, MHz-frequency ultrasound in a tissue-mimicking material.”Ultrasound in medicine & biology 27, no. 10 (2001): 1399-1412.[31] Bouakaz, Ayache, Nico De Jong, and Christian Cachard. ”Standard properties ofultrasound contrast agents.” Ultrasound in medicine & biology 24, no. 3 (1998):469-472.[32] Sojahrood, Amin Jafari, Omar Falou, Robert Earl, Raffi Karshafian, and MichaelC. Kolios. ”Influence of the pressure-dependent resonance frequency on thebifurcation structure and backscattered pressure of ultrasound contrast agents:a numerical investigation.” Nonlinear Dynamics 80, no. 1-2 (2015): 889-904..[33] Segers, Tim, Pieter Kruizinga, Maarten P. Kok, Guillaume Lajoinie, NicoDe Jong, and Michel Versluis. ”Monodisperse versus polydisperse ultrasoundcontrast agents: Non-linear response, sensitivity, and deep tissue imagingpotential.” Ultrasound in medicine & biology 44, no. 7 (2018): 1482-1492.[34] Zderic, Vesna, Jessica Foley, Wenbo Luo, and Shahram Vaezy. ”Prevention ofpostfocal thermal damage by formation of bubbles at the focus during highintensity focused ultrasound therapy.” Medical physics 35, no. 10 (2008): 4292-4299.[35] Soetanto, Kawan, and Man Chan. ”Fundamental studies on contrast images fromdifferent-sized microbubbles: analytical and experimental studies.” Ultrasoundin medicine & biology 26, no. 1 (2000): 81-91.
36] Sojahrood, Amin Jafari, and Michael C. Kolios. ”The utilization of the bubblepressure dependent harmonic resonance frequency for enhanced heating duringhigh intensity focused ultrasound treatments.” In American Institute of PhysicsConference Series, vol. 1481, pp. 345-350. 2012.[37] Commander, Kerry W., and Andrea Prosperetti. ”Linear pressure waves inbubbly liquids: Comparison between theory and experiments.” The Journal ofthe Acoustical Society of America 85, no. 2 (1989): 732-746.[38] Louisnard, Olivier. ”A simple model of ultrasound propagation in a cavitatingliquid. Part I: Theory, nonlinear attenuation and traveling wave generation.”Ultrasonics sonochemistry 19, no. 1 (2012): 56-65.[39] Jamshidi, Rashid, and Gunther Brenner. ”Dissipation of ultrasonic wavepropagation in bubbly liquids considering the effect of compressibility to thefirst order of acoustical Mach number.” Ultrasonics 53, no. 4 (2013): 842-848.[40] Sojahrood, A.J., Haghi, H., Karshfian, R. and Kolios, M.C., 2020. Criticalcorrections to models of nonlinear power dissipation of ultrasonically excitedbubbles. Ultrasonics Sonochemistry, p.105089.[41] Sojahrood, A.J., Haghi, H., Li, Q., Porter, T.M., Karshfian, R. and Kolios, M.C.,2020. Nonlinear power loss in the oscillations of coated and uncoated bubbles:Role of thermal, radiation and encapsulating shell damping at various excitationpressures. Ultrasonics Sonochemistry, p.105070.[42] Hoff, Lars, Per C. Sontum, and Jens M. Hovem. ”Oscillations of polymericmicrobubbles: Effect of the encapsulating shell.” The Journal of the AcousticalSociety of America 107, no. 4 (2000): 2272-2280.[43] Mantouka, Agni, Hakan Dogan, P. R. White, and T. G. Leighton. ”Modellingacoustic scattering, sound speed, and attenuation in gassy soft marine sediments.”The journal of the acoustical society of America 140, no. 1 (2016): 274-282.[44] Dogan, Hakan, Paul R. White, and Timothy G. Leighton. ”Acoustic wavepropagation in gassy porous marine sediments: The rheological and the elasticeffects.” The Journal of the Acoustical Society of America 141, no. 3 (2017):2277-2288.[45] Segers, T. J., N. de Jong, and Michel Versluis. ”Uniform scattering andattenuation of acoustically sorted ultrasound contrast agents: Modeling andexperiments.” Journal of the Acoustical Society of America 140, no. 4 (2016):2506-2517.[46] Dogan, Hakan, and Viktor Popov. ”Numerical simulation of the nonlinearultrasonic pressure wave propagation in a cavitating bubbly liquid inside asonochemical reactor.” Ultrasonics sonochemistry 30 (2016): 87-97.[47] Sojahrood, A. J., Q. Li, H. Haghi, R. Karshafian, T. M. Porter, and M. C. Kolios.”Pressure dependence of the ultrasound attenuation and speed in bubbly media:Theory and experiment.” arXiv preprint arXiv:1811.07788 (2018).
48] Plesset, Milton S. ”The dynamics of cavitation bubbles.” Journal of appliedmechanics 16 (1949): 277-282.[49] Keller, Joseph B., and Michael Miksis. ”Bubble oscillations of large amplitude.”The Journal of the Acoustical Society of America 68, no. 2 (1980): 628-633.[50] Sojahrood, A. J., D. Wegierak, H. Haghi, R. Karshfian, and M. C. Kolios. ”Asimple method to analyze the super-harmonic and ultra-harmonic behavior ofthe acoustically excited bubble oscillator.” Ultrasonics sonochemistry 54 (2019):99.[51] Morgan, Karen E., John S. Allen, Paul A. Dayton, James E. Chomas, A. L.Klibaov, and Katherine W. Ferrara. ”Experimental and theoretical evaluationof microbubble behavior: Effect of transmitted phase and bubble size.” IEEEtransactions on ultrasonics, ferroelectrics, and frequency control 47, no. 6 (2000):1494-1509.[52] Toegel, Ruediger, Bruno Gompf, Rainer Pecha, and Detlef Lohse. ”Does watervapor prevent upscaling sonoluminescence?.” Physical review letters 85, no. 15(2000): 3165.[53] Lide, David R., and Henry V. Kehiaian. CRC handbook of thermophysical andthermochemical data. Crc Press, 1994.[54] http://detector-cooling.web.cern.ch/Detector-Cooling/data/C3F8 − Properties.pdf[55] Stricker, Laura, Andrea Prosperetti, and Detlef Lohse. ”Validation of anapproximate model for the thermal behavior in acoustically driven bubbles.”The Journal of the Acoustical Society of America 130, no. 5 (2011): 3243-3251.[56] Sojahrood, A.J., Wegierak, D., Haghi, H., Karshafian, R. and Kolios, M.C.,2018. A comprehensive bifurcation method to analyze the super-harmonic andultra-harmonic behavior of the acoustically excited bubble oscillator. arXivpreprint arXiv:1810.01239.[57] Flynn, H.G., Church, C.C.: Transient pulsations of small gas bubbles in water.J. Acoust. Soc. Am. 84, 985998 (1988)[58] Sojahrood, A.J., Earl, R., Kolios, M.C. and Karshafian, R., 2020. Investigationof the 1/2 order subharmonic emissions of the period-2 oscillations of anultrasonically excited bubble. Physics Letters A, p.126446.[59] A.J. Sojahrood, R.E. Earl, M.C. Kolios and R. Karshafian, Nonlinear dynamicsof acoustic bubbles excited by their pressure dependent subharmonic resonancefrequency: oversaturation and enhancement of the subharmonic signal, arxiv:2019[60] Sojahrood A.J. & M.C. Classification of the nonlinear dynamics and bifurcationstructure of ultrasound contrast agents excited at higher multiples of theirresonance frequency. Physics Letters A, 376(33), pp.2222-2229.
61] Hegeds, F., Hs, C. and Kullmann, L., 2012. Stable period 1, 2 and 3 structuresof the harmonically excited RayleighPlesset equation applying low ambientpressure. The IMA Journal of Applied Mathematics, 78(6), pp.1179-1195.[62] Hegeds, F., 2016. Topological analysis of the periodic structures in a harmonicallydriven bubble oscillator near Blake’s critical threshold: Infinite sequence of two-sided Farey ordering trees. Physics Letters A, 380(9-10), pp.1012-1022.[63] J.A. Rooney, Hemolysis near an ultrasonically pulsating gas bubble, Science 169(1970) 869871.[64] J. Wu, W.L. Nyborg, Ultrasound, cavitation bubbles and their interaction withcells, Adv. Drug Deliv. Rev. 60 (2008) 11031116.[65] Bouakaz, A., De Jong, N. and Cachard, C., 1998. Standard properties ofultrasound contrast agents. Ultrasound in medicine & biology, 24(3), pp.469-472.[66] Goertz, D.E., de Jong, N. and van der Steen, A.F., 2007. Attenuation and sizedistribution measurements of Definity and manipulated Definity populations.Ultrasound in medicine & biology, 33(9), pp.1376-1388.[67] Raymond, J.L., Haworth, K.J., Bader, K.B., Radhakrishnan, K., Griffin, J.K.,Huang, S.L., McPherson, D.D. and Holland, C.K., 2014. Broadband attenuationmeasurements of phospholipid-shelled ultrasound contrast agents. Ultrasound inmedicine & biology, 40(2), pp.410-421.[68] Shekhar, H., Smith, N.J., Raymond, J.L. and Holland, C.K., 2018. Effect oftemperature on the size distribution, shell properties, and stability of Definity.Ultrasound in medicine & biology, 44(2), pp.434-446.[69] Shekhar, H., Kleven, R.T., Peng, T., Palaniappan, A., Karani, K.B., Huang,S., McPherson, D.D. and Holland, C.K., 2019. In vitro characterization ofsonothrombolysis and echocontrast agents to treat ischemic stroke. Scientificreports, 9(1), p.9902.[70] Helfield, B.L., Leung, B.Y., Huo, X. and Goertz, D.E., 2014. Scaling of theviscoelastic shell properties of phospholipid encapsulated microbubbles withultrasound frequency. Ultrasonics, 54(6), pp.1419-1424.[71] Xia, L., Porter, T.M. and Sarkar, K., 2015. Interpreting attenuation at differentexcitation amplitudes to estimate strain-dependent interfacial rheologicalproperties of lipid-coated monodisperse microbubbles. The Journal of theAcoustical Society of America, 138(6), pp.3994-4003.[72] Bouakaz, A., Frigstad, S., Ten Cate, F.J. and de Jong, N., 2002. Super harmonicimaging: a new imaging technique for improved contrast detection. Ultrasoundin medicine & biology, 28(1), pp.59-68.[73] Cherin, E., Yin, J., Forbrich, A., White, C., Dayton, P.A., Foster, F.S. andDmor, C.E., 2019. In Vitro Superharmonic Contrast Imaging Using a HybridDual-Frequency Probe. Ultrasound in medicine & biology. 45(9), pp.2525-2539.
74] Perera, R., Hernandez, C., Cooley, M., Jung, O., Jeganathan, S., Abenojar,E., Fishbein, G., Sojahrood, A.J., Emerson, C., Stewart, P.L. and Kolios, M.,2019. Contrast Enhanced Ultrasound Imaging by Nature-Inspired UltrastableEchogenic Nanobubbles. Nanoscale.[75] Shekhar, H., Rowan, J.S. and Doyley, M.M., 2017. Combining subharmonicand ultraharmonic modes for intravascular ultrasound imaging: A preliminaryevaluation. Ultrasound in medicine & biology, 43(11), pp.2725-2732.[76] De Jong, N., Emmer, M., Van Wamel, A. and Versluis, M., 2009. Ultrasoniccharacterization of ultrasound contrast agents. Medical & biological engineering& computing, 47(8), pp.861-873.[77] Haworth, K.J., Salgaonkar, V.A., Corregan, N.M., Holland, C.K. and Mast,T.D., 2015. Using passive cavitation images to classify high-intensity focusedultrasound lesions. Ultrasound in medicine & biology, 41(9), pp.2420-2434.[78] Alli, S., Figueiredo, C.A., Golbourn, B., Sabha, N., Wu, M.Y., Bondoc, A., Luck,A., Coluccia, D., Maslink, C., Smith, C. and Wurdak, H., 2018. Brainstem bloodbrain barrier disruption using focused ultrasound: A demonstration of feasibilityand enhanced doxorubicin delivery. Journal of controlled release, 281, pp.29-41.[79] Eckersley, R.J., Chin, C.T. and Burns, P.N., 2005. Optimising phase andamplitude modulation schemes for imaging microbubble contrast agents atlow acoustic power. Ultrasound in medicine & biology, 31(2), pp.213-219.[80] Phillips, P.J., 2001, October. Contrast pulse sequences (CPS): imagingnonlinear microbubbles. In 2001 IEEE Ultrasonics Symposium. Proceedings. AnInternational Symposium (Cat. No. 01CH37263) (Vol. 2, pp. 1739-1745). IEEE.[81] Goertz, D.E., Frijlink, M.E., Tempel, D., Bhagwandas, V., Gisolf, A., Krams, R.,de Jong, N. and van der Steen, A.F., 2007. Subharmonic contrast intravascularultrasound for vasa vasorum imaging. Ultrasound in medicine & biology, 33(12),pp.1859-1872.[82] Goertz, D.E., Cherin, E., Needles, A., Karshafian, R., Brown, A.S., Burns, P.N.and Foster, F.S., 2005. High frequency nonlinear B-scan imaging of microbubblecontrast agents. IEEE transactions on ultrasonics, ferroelectrics, and frequencycontrol, 52(1), pp.65-79.[83] Forsberg, F., Piccoli, C.W., Merton, D.A., Palazzo, J.J. and Hall, A.L., 2007.Breast lesions: imaging with contrast-enhanced subharmonic USinitial experience.Radiology, 244(3), pp.718-726.[84] M.A. OReilly, et al, Focused-ultrasound disruption of the blood-brain barrierusing closely-timed short pulses: inflence of sonication parameters and injectionrate, Ultrasound in medicine & biology 37 (2011): 587-594.[85] Haworth, K.J., Bader, K.B., Rich, K.T., Holland, C.K. and Mast, T.D., 2016.Quantitative frequency-domain passive cavitation imaging. IEEE transactionson ultrasonics, ferroelectrics, and frequency control, 64(1), pp.177-191.
86] Eller, Anthony, and H. G. Flynn. ”Generation of subharmonics of order onehalfby bubbles in a sound field.” The Journal of the Acoustical Society of America46.3B (1969): 722-727.[87] Prosperetti, Andrea. ”Nonlinear oscillations of gas bubbles in liquids: steadystatesolutions.” The Journal of the Acoustical Society of America 56.3 (1974): 878-885.[88] Prosperetti, Andrea. ”Application of the subharmonic threshold to themeasurement of the damping of oscillating gas bubbles.” The Journal of theAcoustical Society of America61.1 (1977): 11-16.[89] Prosperetti, Andrea. ”A general derivation of the subharmonic threshold fornon-linear bubble oscillations.” The Journal of the Acoustical Society of America133.6 (2013): 3719-3726[90] P. D., P. M. Shankar, and V. L. Newhouse. ”Subharmonic generation fromultrasonic contrast agents.” Physics in Medicine & Biology 44.3 (1999): 681.[91] Shankar, P. M., P. D. Krishna, and V. L. Newhouse. ”Subharmonic backscatteringfrom ultrasound contrast agents.” The Journal of the Acoustical Society ofAmerica106.4 (1999): 2104-2110.[92] Katiyar, Amit, and Kausik Sarkar. ”Effects of encapsulation damping on theexcitation threshold for subharmonic generation from contrast microbubbles.”The Journal of the Acoustical Society of America 132.5 (2012): 3576-3585.[93] Katiyar, Amit, and Kausik Sarkar. ”Excitation threshold for subharmonicgeneration from contrast microbubbles.” The Journal of the Acoustical Societyof America 130.5 (2011): 3137-3147.[94] Jafari Sojahrood, A. and Kolios, M., 2011. Theoretical considerations forultrasound contrast agent amplitude modulation techniques at high frequencies.The Journal of the Acoustical Society of America, 129(4), pp.2511-2511.[95] Wang, M. and Zhou, Y., 2018. Numerical investigation of the inertial cavitationthreshold by dual-frequency excitation in the fluid and tissue. Ultrasonicssonochemistry, 42, pp.327-338.[96] Kooiman, K., Vos, H.J., Versluis, M. and de Jong, N., 2014. Acoustic behaviorof microbubbles and implications for drug delivery. Advanced drug deliveryreviews, 72, pp.28-48.[97] T.J. Mason, Ultrasonic cleaning: a historical perspective, Ultrason. Sonochem.29 (2016) 519523.[98] Maisonhaute, E., Prado, C., White, P.C. and Compton, R.G., 2002. Surfaceacoustic cavitation understood via nanosecond electrochemistry. Part III: Shearstress in ultrasonic cleaning. Ultrasonics sonochemistry, 9(6), pp.297-303.[99] Yamashita, T. and Ando, K., 2019. Low-intensity ultrasound induced cavitationand streaming in oxygen-supersaturated water: Role of cavitation bubbles asphysical cleaning agents. Ultrasonics sonochemistry, 52, pp.268-279. Pressure (Pa) R / R R0=2 m uncoated bubble,f=0.3*fr
Pressure (Pa) -22 -20 -18 -16 -14 D i ss i p a t e d po w e r ( W ) R0=2 m,f=0.3*fr
TdRdLd (a) (b)
Pressure (Pa) -22 -20 -18 -16 D i ss i p a t e d po w e r ( W ) R0=2 m,f=0.5*fr
TdRdLd (c) (d)
Pressure (Pa) -20 -19 -18 -17 D i ss i p a t e d po w e r ( W ) R0=2 m,f=0.9*fr
TdRdLd (e) (f)
Pressure (Pa) -20 -18 -16 D i ss i p a t e d po w e r ( W ) R0=2 m,f=1*fr
TdRdLd (g) (h)
Fig. A.1. Bifurcation structure (left column) and the dissipated powers (right col-umn) of the oscillations of an uncoated air bubble with R = 2 µm for f = 0 . f r (a-b)- f = 0 . f r (c-d)- f = 0 . f r (e-f) & f = f r (g-h). Bifurcation structure of an uncoated Air bubble with R = 2 µm and a coated bubble with R = 4 µm A.1 The case of an uncoated air bubble with R = 2 µm The same nonlinear oscillations regimes are also studied for an uncoated airbubble with R = 2 µm . Due to the smaller size of the bubble, a larger contri-bution from viscous effects and a smaller contribution for thermal effects areexpected.Fig. A1.a shows the case of sonication with f = 0 . f r . P1 oscillations undergoa 3rd SuH resonance at P a (cid:117) kP a (P1 oscillations with 3 maxima). At P a (cid:117) kP a , the amplitude of one of the maxima coincides with the amplitudeof the red curve indicating that the wall velocity becomes in phase with thedriving pressure every acoustic cycle. At P a (cid:117) kP a the red curve undergoesa Pd and P1 oscillations become P2. At the same time Pd occurs in the bluecurve and UH oscillations of 7/2 order develop. The red curve grows fast withincreasing pressure and with the occurrence of 3rd order SuH and Pd, theamplitude of the red curve becomes the same as the highest amplitude of theblue curve right when Pd takes place, indicating wall velocity is in phase withthe excitation pressure. During UH oscillations the value of the upper andlower branch of P2 oscillations in red curve are exactly the same as the twohighest amplitudes of the blue curve. A small chaotic window appears at 170kPa and then a 2nd order SuH giant (P1 with 2 maxima) resonance emergesout of the chaotic window which later undergoes successive Pds to chaos. Dueto the smaller bubble size compared to the 10 µm bubble (analyzed in Fig.1b), Ld and Td have the same value for pressures below 120 kPa and Td isno longer the dominant power dissipation mechanism. Rd grows faster withincreasing pressure and the contributions of Td, Ld and Rd become similarwhen Pd takes place. Rd > Ld > T d for the UH regime of oscillations. Thegiant resonance is concomitant with a sharper increase in Rd making Rd thedominant damping mechanism during giant resonant oscillations.When f = 0 . f r (Fig. A.1c) oscillations are of P1 with one maximum for P a < kP a ; 2nd order SuH oscillations occur at (cid:117) kP a (2 maxima appearin the blue curve). At P a = 90 kP a , the amplitude of the red curve becomesequal to the highest amplitude maximum of the blue curve. This is concomitantwith the saturation of 2nd order SuH frequency component of the scatteredpressure ( P sc ). At 127 kPa, the red curve undergoes a Pd concomitant witha Pd in the blue curve resulting in a P2 oscillation with 4 maxima (3 / (cid:117)
140 kPa
R/R = 2, and the bubble possibly can notsustain non-destructive oscillations beyond this pressure. For P a > kP a achaotic window emerges and later at P a (cid:117) kP a a giant period one reso-nance emerges out of the chaotic window. Fig. A.1d shows the correspondingdissipated powers. Ld and Rd are approximately equal for pressures below44he occurrence of Pd and UHs. Simultaneous with the 3 / Rd > Ld > T d . Rd undergoes the sharpestincrease concomitant with the generation of giant resonance, making it thestrongest dissipation mechanism at higher pressures.When f = 0 . f r (Fig. A.1e), which is a P Df r [32], the oscillations are of P1and grow monotonically with increasing pressure and at P a (cid:117) kP a (thepressure of the P Df r ) the oscillation amplitude undergo a sharp increase withthe red and blue curve coinciding with each other. Above P a (cid:117) kP a , R/R exceeds 2 (black horizontal curve) and beyond this point bubble destructionis likely. Oscillations undergo Pd at (cid:117)
220 kPa and a small chaotic windowoccurs at P a (cid:117) kP a through successive Pds. At (cid:117)
275 kPa a P3 oscillationswith 3 maxima emerges out of the chaotic window until P a (cid:117)
350 kPa wherechaos is regenerated. The power dissipation graph in Fig. A.1f indicates that
T d > Ld > Rd before the SN bifurcation takes place. Above P a (cid:117) kP a (pressure for SN bifurcation) Ld becomes stronger and the dissipation orderis Ld > T d > Rd . After SN, Rd grows faster with pressure increase while Tdstays constant. Rd supersedes Td at P a (cid:117)
125 kPa and becomes equal to Ldwhen Pd takes place. Ld, Rd and Td then stay relatively constant for theP2 oscillations regimes (this can be due to the decrease in wall velocity andacceleration when Pd takes place in cases where the bubble is sonicated witha frequency near f = f r [32]). Emergence of the P3 oscillations out of thechaotic window with high amplitude is concomitant with an increase in Rdand decrease in Ld and Td; this will lead to an increase in the STDR, however,with the possible trade off the loss of stable oscillations.When f = f r (Fig. A.1g); at lower pressures ( P a < kP a ) oscillations areof P1 and the wall velocity is in phase with the driving acoustic force (blueand red curve are on top of each other) indicating resonant oscillations. Fur-ther pressure increase results in possible bubble destruction at P a = 120 kP a (black horizontal line meets the blue line). At P a (cid:117) kP a , occurrence of Pdresults in P2 oscillations for 220 < P a <
260 followed by the emergence of asmall chaotic window through successive Pds. Similar to the previous case,a P3 emerges from the chaotic window followed by regeneration of chaos for320 kP a < P a < kP a . At (cid:117) kP a a giant P2 resonance emerges out ofthe chaotic window. The corresponding dissipated powers illustrated in Fig.A.1h show that for pressures below ≈ kP a , T d > Ld > Rd . Above thispressure
Ld > T d > Rd until the excitation pressure reaches (cid:117)
190 kPa andRd becomes equal to Td. When Pd occurs, Rd overcomes Td; then Rd, Ldand Td stay constant during P2 oscillations. Generation of P4 results in anincrease in Rd and a subsequent decrease in Ld and Td. Emergence of P3results in an increase in Rd making the order as
Rd > Ld > T d followed bya sharp increase of Rd when giant resonance takes place. Similar to previouscases Td decreases when giant resonance occurs, and Ld increases howeverwith a smaller percentage compared to Rd. Once again, the giant resonancecan lead to a significant increase in STDR; however this may lead to bubbledestruction. The generation of higher temperatures due to stronger collapses45nd the decrease in Td may have consequences in enhancing chemical reactionswithin the bubble.The case of f = 1 . f r is shown in Fig. A.2a. Oscillation amplitude increasesmonotonically with pressure and bubble undergoes Pd at (cid:117) kP a . Unlikethe case of sonication with f = f r , (and similar to the uncoated air bubblewith R = 10 µm ) Pd occurs when R/R < (cid:117) kP a oscillations become chaotic throughsuccessive Pds and chaos stretches until P a (cid:117) kP a . At this pressure a P2giant resonant oscillation emerges out of the chaotic window (the solution withthe higher amplitude in red curve is exactly equal to the smaller maximum in-dicating wall velocity becomes in phase with the driving acoustic pressure onceevery two acoustic cycles). The P2 giant resonance undergoes successive Pdsto chaos at (cid:117) kP a . The corresponding dissipated power graphs (Fig A.2b)show that Ld (cid:117) T d > Rd for P a (cid:46)
225 kPa. When Pd occurs, Ld becomesstronger than Td with
Ld > T d > Rd ; once again, during P2 oscillations,Ld, Td and Rd stay relatively constant as pressure increases. Generation ofP4 and chaos is concomitant with an increase in Rd making Ld (cid:117) Rd > T d .This is similar to the previous cases when the giant resonance emerges Rdand Ld undergo a sharp increase (Rd exhibits the sharpest increase), whileTd decreases slightly. This makes the contribution order of the dissipationmechanisms as
Rd > Ld > T d .When f = 1 . f r (the P Df sh [59]) the P1 oscillation amplitude grows slowlywith increasing pressure and the bubble undergoes a Pd at (cid:117) kP a . Genera-tion of Pd is concomitant with a sharp increase in the oscillation amplitude(oscillations are P2 and have two maxima). At 200 kPa P2 oscillations un-dergo a SN bifurcation to P2 oscillations of higher amplitude (here the signallooses one of its maxima [59]). As pressure increases the second maxima isgenerated at (cid:117) kP a . At 300 kPa, R/R becomes larger than 2 (black line).P2 oscillations undergo period doubling to P4-2 oscillations and a chaoticwindow appears at 405 kPa through successive Pds of the P4-2 signal. Laterat 440 kPa, a P6 oscillation with 6 maxima emerges out of the chaotic windowwhich through successive Pds translate to P12 and chaos at (cid:117) − kP a .The corresponding power graphs (Fig. A.2d) show that T d = Ld > Rd below P a (cid:117) kP a where Pd takes place. Generation of Pd results in a decrease inTd and Ld becomes stronger than Td. Simultaneous with the SN bifurcationat P a = 200 kP a , Rd, Ld and Td undergo a sharp increase (with Rd exhibitingthe highest increase). This makes Rd approximately equal to Ld and for therest of the P2 oscillations, power dissipation stays relatively constant withincreasing pressure and Rd = Ld > T d . Generation of chaos results in somesporadic fluctuations and when P6 emerges out of chaos Ld and Td decreaseresulting in
Rd > Ld > T d .When f = f sh ( f = 2 f r in Fig. A.2e), P1 oscillations slowly grow with in-creasing pressure and at P a (cid:117) kP a , a Pd takes place, and concomitantwith Pd, oscillation amplitude start growing quickly. P2 oscillations evolve inthe form of a bow-tie. Right when Pd occurs, oscillations have two maxima,46 Pressure (Pa) -22 -20 -18 -16 D i ss i p a t e d po w e r ( W ) R0=2 m,f=1.2*fr
TdRdLd (a) (b)
Pressure (Pa) -22 -20 -18 D i ss i p a t e d po w e r ( W ) R0=2 m,f=1.85*fr
TdRdLd (c) (d)
Pressure (Pa) -22 -20 -18 -16 D i ss i p a t e d po w e r ( W ) R0=2 m,f=2*fr
TdRdLd (e) (f)
Pressure (Pa) -22 -20 -18 D i ss i p a t e d po w e r ( W ) R0=2 m,f=3*fr
TdRdLd (g) (h)
Fig. A.2. Bifurcation structure (left column) and the dissipated power (right col-umn) of the oscillations of an uncoated air bubble with R = 2 µm for f = 1 . f r (a-b)- f = 1 . f r (c-d)- f = 2 f r (e-f) & f = 3 f r (g-h). (cid:117) kP a (in [58] we haveshown that this may be the point where 1 / P sc gets saturated). Oscillations undergo Pds at 395 kPa to P4-2 oscillationsand when P a (cid:117) kP a , R/R exceeds 2 (black horizontal line collides withthe blue curve). A small window of P6-2 ([58]) occurs right before 400 kPa.Later chaos appears at (cid:117) kP a . For P a < kP a , T d = Ld = Rd andsimultaneous with Pd, dissipation powers undergo a fast increase; but, theyquickly plateau with pressure increase. Further increase in pressure results ina slight decrease in Td and a slight increase in Rd and Ld. At P a (cid:117) kP a where the amplitude of the chaotic oscillations sharply increases; Rd becomesstronger than Ld.When f = 3 f r (Fig. A.2g), oscillations grow very slowly with pressure increaseuntil at (cid:117) kP a at which P1 oscillations undergo a SN bifurcation to P3 os-cillations with 2 maxima. Oscillation amplitude increase slowly with increasingpressure and a small P6 window appears at ≈ P a (cid:117) kP a . The corresponding dissipated power graphs (Fig.A.2h) show that similar to the case of unacoated air bubble with R = 10 µm , Rd > Ld > T d before the occurrence of the SN bifurcation. Occurrence ofSN bifurcation is concomitant with a sharp increase in Rd and Ld and Tdwith (
Rd > Ld > T d ). As pressure increases the difference between Rd, Ldand Td diverges resulting in an increase in STDR. A return to P1 oscillationsis concomitant with a decrease in the dissipation. Due to the larger increasein Td when P3 occurs, the STDR decreases. The increase in Td is due tothe large average surface area of the bubbles and a slower rebound during P3oscillations.
A.2 The case of a coated C3F8 bubble with R = 4 µm In this subsection we analyze the dynamics of the coated bubble with R = 4 µm with a C3F8 gas core. Fig. A3a displays the bifurcation structure of a coatedbubble with R = 4 µm when f = 0 . f r . The oscillation amplitude grows withpressure and at P a = 80 kP a , 3 maxima appear in the P1 oscillations (3rdorder SuH [50]). The 3rd order SuH undergoes a SN bifurcation to another3rd order SuH oscillations at P a (cid:117) kP a . The signal is still P1 with 3maxima and at P a (cid:117) kP a Pd takes place and 7 / P a = 200 kP a followed by a P1 signal of 2nd order SuH, then 5 / Cd > Ld > T d for P a < kP a . When SuH oscillations48 Pressure (Pa) -20 -18 -16 -14 D i ss i p a t e d po w e r ( W ) R0=4 m coated bubble,f=0.25*fr
TdRdLdCd (a) (b)
Pressure (Pa) -20 -18 -16 -14 D i ss i p a t e d po w e r ( W ) R0=4 m coated bubble,f=0.45*fr
TdRdLdCd (c) (d)
Pressure (Pa) -20 -18 -16 D i ss i p a t e d po w e r ( W ) R0=4 m coated bubble,f=0.8*fr
TdRdLdCd (e) (f)
Pressure (Pa) -20 -18 -16 D i ss i p a t e d po w e r ( W ) R0=4 m coated bubble,f=1*fr
TdRdLdCd (g) (h)
Fig. A.3. Bifurcation structure (left column) and the dissipated power as a functionof pressure (right column) of the oscillations of a coated C3F8 bubble with R = 4 µm for f = 0 . f r (a-b)- f = 0 . f r (c-d)- f = 0 . f r (e-f) & f = f r (g-h). f = 0 . f r . As pressure increases,the P1 oscillation amplitude increases and at (cid:117) kP a two maxima appearin the P1 oscillations. At approximately 105 kPa; bubble collapses two timesin each acoustic cycles with the wall velocity of one of the collapses in phasewith the driving acoustic pressure (red and blue line have the same value).Pd takes place at (cid:117) kP a (5 / R/R becomes equal to 2 at the pressure at which chaosis generated at P a (cid:117) kP a . Slightly below 200 kPa a giant P1 resonanceemerges out of the chaotic window which later undergoes successive Pds tochaos at P a (cid:117) kP a . The corresponding power dissipation curves in Fig.A3d shows that for pressures below the SuH oscillations, Cd > Ld > T d (cid:117) Rd .After the generation of SuH oscillations Rd supersedes Td and becomes equalto Ld at about 105 kPa (when red curve meets the blue curve in Fig. A3c).Power dissipation curves plateau and when Pd occurs. Rd, Cd, Ld and Tdincrease. During 5 / Cd > Rd > Ld > T d . Emergence of giantresonance is concomitant with an increase in all the dissipation mechanismswith Rd exhibiting the sharpest growth. Afterwards Rd grows faster and be-comes approximately equal to Cd during chaotic oscillations. Similar to Fig.A3b, Rd and Cd are 3 orders of magnitude larger than Td and about an orderof magnitude stronger than Ld.When f = 0 . f r ( P Df r [32]) a SN bifurcation takes place at P a = 80 kP a andthe oscillation amplitude R/R exceeds at P a (cid:117) kP a (black line meets theblue curve). At (cid:117) kP a , Pd takes place and afterwards P2 oscillations un-dergo successive period doubling to chaos at (cid:117) kP a . A P2 giant resonancewith 2 maxima emerges out of the chaotic window at 520 kPa. The correspond-ing dissipation curves in Fig. A.3f reveal that when the SN occurs, Cd, Ld,Rd and Td undergo a sharp increase with Td exhibiting the smallest increase.Before the SN, Cd > Ld > Rd > T d and after the SN,
Cd > Ld = Rd > T d .When
R/R exceeds 2, Rd becomes stronger than Ld and power dissipationcontribution is in the following order for the rest of the pressure range thatstudied: Cd > Rd > Ld > T d . Emergence of the giant resonance is simultane-ous with an increase in Rd, Cd and Ld (with Rd demonstrating the largestincrease) and Td decreases.When f = f r (Fig. A.3g) the oscillation amplitude grows monotonically withpressure. For excitation pressures below 50 kPa, the red curve and blue curveshave the same value (wall velocity is in phase with the driving acoustic pres-sure). The two curves diverge as pressure increases and at 230 kPa Pd takesplace. P2 oscillations amplitudes exceed R/R = 2 at 260 kPa; afterwards50uccessive Pds take place in the bifurcation structure resulting in chaotic oscil-lations at ≈ kP a . The chaotic window continues up to 420 kPa; whereby,large amplitude P2 oscillations emerge out of the chaotic window (one of thered solutions is equal to the smallest maxima in blue curve). Chaos is thengenerated through successive Pds at 520 kPa. Power dissipation curves in Fig.A.3h show that Cd > Rd (cid:117)
Ld > T d ; however, when Pd occurs, Rd slightlyexceeds Ld due to the fact that both Cd and Ld undergo a slightly higherdecrease compared to Rd. This is possibly due to decrease in the wall velocityconcomitant with Pd when bubble is sonicated with a frequency close to itsresonance frequency [32]. Emergence of the giant P2 oscillations is concomitantwith a very sharp increase in Rd and Cd, a slight increase in Ld and minimalchanges in Td.When f = 1 . f r (Fig. A.4a), the oscillation amplitude grows monotonicallywith pressure. A Pd takes place at (cid:117) kP a . Bubbling bifurcation takes placein each of the branches of the P2 regime and a small window of chaos appearsfollowed by a small P3 window which undergoes Pd to P12 after which thereis an interesting reverse Pd leading to a sudden onset of chaos. The oscillationamplitude exceeds R/R = 2 at (cid:117) kP a . The chaotic window extends until P a = 1 M P a where a giant P3 emerges out of the chaotic window which thenundergoes Pd to P12 oscillations. The corresponding dissipation curves in Fig.A4b show that
Cd > Ld = Rd > T d until Pd after which Rd becomes slightlylarger than Ld. Pd results in a notable decrease in Cd and Ld. Td remainstwo orders of magnitude less than Cd (Fig. A.3b). When the first P3 oscilla-tion occurs, Cd, Rd and Ld undergo a sharp increase with the most notableincrease in Rd. Reverse Pd bifurcation results in a decrease in dissipation dueto lower oscillation amplitudes. Initiation of chaos leads to a sharp increase indissipation with
Cd > Rd > Ld > T d . Rd grows faster than other mechanismsas pressure increases and becomes equal to Cd at ≈ kP a . Finally whenthe P3 giant resonance occurs, Cd, Rd and Ld undergo an increase with Rdexperiencing the largest growth. Td undergoes a small decrease during P3giant resonance oscillations.Fig. A.4c represents the case of sonication with f = 1 . f r ( P Df sh [59]). P1oscillations grow slowly with pressure and at (cid:117) kP a a SN bifurcation fromP1 oscillations of lower amplitude to P2 oscillations (with one maximum) ofhigher amplitude takes place. Second maxima emerges at (cid:117) kP a ; afterwardsoscillations undergo Pd at (cid:117) kP a which are then followed by successive Pdsto chaos at (cid:117) kP a . Chaotic window stretches until (cid:117) kP a with a smallwindow of P6 oscillations. A P6 oscillation regime with high amplitude emergesout of the chaotic window; later undergoing Pds to P12 and then chaos. Thecorresponding dissipation curves are shown in Fig. A.4d. For pressures belowthe SN bifurcation Cd > Rd > Ld > T d . Occurrence of SN bifurcation is con-comitant with a sharp increase in the dissipated powers.
Cd > Rd > Ld > T d until at higher pressures ( > f = 2 f r (Fig. A.4e) is the f sh of the bubble [58]. P1 oscillations undergo Pd at (cid:117) kP a . The P2 oscillations loose one maxima right after the generation of51 Pressure (Pa) R / R R0=4 m,f=1.2*fr conventional methodmaxima method
Pressure (Pa) -20 -18 -16 D i ss i p a t e d po w e r ( W ) R0=4 m coated bubble,f=1.2*fr
TdRdLdCd (a) (b)
Pressure (Pa) R / R R0=4 m coated bubble,f=1.6*fr conventional methodmaxima method
Pressure (Pa) -22 -20 -18 -16 D i ss i p a t e d po w e r ( W ) R0=4 m coated bubble,f=1.6*fr
TdRdLdCd (c) (d)
Pressure (Pa) R / R R0=4 m coated, f=2*fr conventional methodmaxima method
Pressure (Pa) -24 -22 -20 -18 -16 -14 D i ss i p a t e d po w e r ( W ) R0=4 m coated bubble,f=2*fr
TdRdLdCd (e) (f)
Pressure (Pa) -24 -22 -20 -18 D i ss i p a t e d po w e r ( W ) R0=4 m coated bubble,f=3*fr
TdRdLdCd (g) (h)
Fig. A.4. Bifurcation structure (left column) and the dissipated powers (right col-umn) of the oscillations of a coated C3F8 bubble with R = 4 µm for f = 1 . f r (a-b)- f = 1 . f r (c-d)- f = 2 f r (e-f) & f = 3 f r (g-h).
52d and then evolve in a form of a bow-tie with the second maxima re-emergingwith an amplitude equal to the larger branch of the red curve (cid:117) kP a .Consistent with previous observations in [58], sonication with f = 2 f r resultsin the largest pressure range of stable P2 oscillations. Oscillations undergoPd to P4 oscillations followed by a SN bifurcation to P4 oscillations of higheramplitude at (cid:117) kP a ; before successive Pds to chaos. Amplitude of thechaotic oscillations increases at (cid:117) kP a which can possibly lead to bubbledestruction as R/R >
2. Chaos continues until 1.1 MPa where a P6 oscillationof large amplitude emerges out of chaos which later undergoes successive Pdsto P12 and chaos. The corresponding dissipation curves are depicted in Fig.A.4f.
Cd > Rd > Ld > T d with dissipated powers undergoing a fast growthconcomitant with Pd. When chaos appears Rd becomes equal to Cd and laterat 1.5 MPa onward
Rd > Cd > Ld > T d . When f = 2 f r , Td is in average twoorders of magnitude smaller than Cd and Rd.The case of sonication with f = 3 f r is shown if Fig. A.4g. Oscillations growslightly with pressure and at (cid:117) kP a a SN bifurcation takes place and P1oscillations turn into P3 oscillations of higher amplitude. P3 then grows inamplitude until it turns to P1 oscillations for a small pressure window andthen again re-emerges through a SN bifurcation. P3 oscillations undergo Pdsto P6 and then return to P1 oscillations for P a (cid:117) . M P a . Correspond-ing dissipated power curves are shown in Fig. A.4h. Unlike previous caseshere
Rd > Cd > Ld > T d . The SN bifurcation results in a large increasein dissipated powers specially in case of Td. This is because during theseP3 oscillations the bubble collapses 3 times out of which two are very gentleand thus a large average bubble radius is maintained during oscillations. Thisincreases the surface area for the heat exchange and Td increases. Moreover,bubble collapses strongly only once in every three cycles; thus high velocity andre-radiated pressure are achieved only once in every three acoustic cycles. Thisis why the average for Ld, Cd and Rd are small. During P3 and P6 oscillations
Cd > Rd > Ld > T d and elsewhere
Rd > Cd > Ld > T d . A.3 Concluding graphs of | ˙ R ( t ) | max , | P sc | max , total dissipated power andSTDR Figure A.5 represents the uncoated air bubble with R = 2 µm . The exactsame behavior of the case of an uncoated bubble with R = 10 µm (Fig. 5)is observed here. Maximum non-destructive V m and P m occurs for f = 0 . f r (2nd SuH) and f = 3 f r respectively. Maximum Rd and W total are achievedwhen f = 0 . f r (3rd SuH). STDR is higher for higher frequencies with themaximum at f = 3 f r . Similar to Fig. 5e and Fig. 6e, the onset of nonlinearoscillations results in a decrease and then increase in STDR if the bubble issonicated above resonance. 53 M ax i m u m b acksca tt e r e d a m p li t ud e ( P a ) R0=2 m uncoated bubble (a) (b) R d ( W ) -17 R0=2 m uncoated bubble f=0.3*frf=0.5*frf=0.9*frf=1*frf=1.2*frf=1.85*frf=2*frf=3*fr t o t a l d i ss i p a t e d po w e r ( W ) -17 R0=2 m uncoated bubble f=0.3*frf=0.5*frf=0.9*frf=1*frf=1.2*frf=1.85*frf=2*frf=3*fr (c) (d) S T DR R0=2 m uncoated bubble f=0.3*frf=0.5*frf=0.9*frf=1*frf=1.2*frf=1.85*frf=2*frf=3*fr (e)
Fig. A.5. Nondestructive (
R/R ≤
2) values of: a) | ˙ R ( t ) | max ( V m ), b) Maximumbackscattered pressure ( | P sc | max ( P m ), c) Rd, d) W total and e) STDR as a functionof pressure in the oscillations of an uncoated air bubble with R = 2 µm . The case of the C3F8 coated bubble with R = 4 µm is shown in Fig. A.6.The same conclusions can be drawn as the two previous cases (Figs. 5, 6 andA.5). This indicates a universal behavior of these parameters in the studiedcases in this paper. 54 V m ( m / s ) R0=4 m coated bubble M ax i m u m b acsca tt e r e d a m p li t ud e ( P a ) R0=4 m coated bubble (a) (b) R d ( W ) -16 R0=4 m coated bubble f=0.3*frf=0.45*frf=0.8*frf=1*frf=1.2*frf=1.4*frf=2*frf=3*fr t o t a l d i ss i p a t e d po w e r ( W ) -15 R0=4 m coated bubble f=0.3*frf=0.45*frf=0.8*frf=1*frf=1.2*frf=1.6*frf=2*frf=3*fr (c) (d) S T DR R0=4 m coated bubble (e)
Fig. A.6. Nondestructive (
R/R ≤
2) values of: a) | ˙ R ( t ) | max ( V m ), b) Maximumbackscattered pressure ( | P sc | max ( P m ), c) Rd, d) W total and e) STDR as a functionof pressure in the oscillations of a coated C3F8 bubble with R = 4 µm ..