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Nonlinear transitions of a spherical cavitation bubble
Chaos, Solitons and Fractals 41 (2009) 818–828
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Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
Nonlinear transitions of a spherical cavitation bubble Sohrab Behnia a,*, Amin Jafari b, Wiria Soltanpoor b, Okhtay Jahanbakhsh b a b
Department of Physics, IAU, Urmia, Iran Department of Physics, Urmia University, Urmia, Iran
a r t i c l e
i n f o
Article history: Accepted 31 March 2008
a b s t r a c t Dynamics of acoustically driven bubbles are known to be complex and uncontrollable. Depending on the applied pressure, frequency and the properties of the bubble and the host media, the radial oscillations of the bubble has been reported to be stable and chaotic. In this paper, the dynamics of a single acoustically driven bubble is widely studied applying the methods of chaos physics. The stability of the bubble is analyzed through plotting the bifurcation diagrams and Lyapunov exponent spectra versus pressure, frequency, initial bubble radius, surface tension and viscosity as the control parameters. Results show the rich nonlinear dynamics of the bubble including period doubling, inverse period doubling, saddle node bifurcations, quasi-periodicity and chaos. Also similarities were detected in the bubble response to variations between some of the control parameters. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction A gas bubble driven in motion by ultrasound is an example of a system with highly nonlinear properties in which deterministic chaos manifests itself. This phenomenon occurs when a high amplitude, high frequency sound is applied to the liquid [1–3]. While high intensity ultrasound is introduced to an aqueous solution, the rare faction cycle may exceed the attractive forces among liquid molecules making cavitation bubbles occur and begin their nonlinear oscillations which is accompanied by intense noise emission; the acoustic cavitation noise [4]. The noise is found not to be caused by statistical rupture process in the liquid but to be of deterministic origin [5] belonging to the class of deterministically chaotic systems [4]. From theoretical point of view, this phenomenon even in its simplest forms is described by a highly nonlinear differential equation of second order [6–9]. Experimentally Esche [10] was the first one to observe subharmonics in the spectrum of the response of liquid in 1952. More after Lauterborn and Cramer [5] observed the subharmonics route to chaos including period doubling bifurcations up to f =8 and reverse bifurcations in the acoustic cavitation noise. Later the bifurcation structure of cavitation was studied numerically and a superstructure resulting from nonlinear resonance was found with period doubling Feigenbaum direct cascades [11] and Grossmann inverse cascades [12] as fine structure. Holographic observations of period doubling and chaotic bubble oscillations in [13] confirmed the view that nonlinear acoustic response of the liquid is mediated by the bubbles and their nonlinear oscillations. The phenomenon of chaotic bubble oscillations was also observed by measuring the time delays between flashes of emitted light in an experiment on sonoluminescence showing period doubling, quasiperiodicity and chaos [14]. The rich nonlinear dynamics of the system was also numerically investigated through applying the methods of chaos physics [15–19]. Besides its complex and nonlinear behavior, acoustic cavitation gas bubbles are present inevitably in lots of applications including material science, sonoluminescence and sonochemistry [20–24], sonofusion [25] and medical procedures such as: lithotripsy, diagnostic imaging, drug and gene delivery, increasing membrane permeability, opening blood brain barrier and
* Corresponding author. E-mail address: [email protected] (S. Behnia). 0960-0779/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2008.04.011
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high intensity focused ultrasound surgery [26–30]. A complete understanding of the dynamics of the phenomena is deemed necessary in order to achieve their full potential use in industry and medicine. For this purpose it is first needed to expand our knowledge on a single bubble behavior under different possible conditions including the frequency and pressure variations of the sound field, fluid and gas properties such as viscosity and surface tension and bubble properties like its initial radius. The results can help us choose a regime to avoid chaotic motion because when a system falls into chaos its behavior is difficult to predict and control [31,32]. In this paper the effects of parameters mentioned above are studied for a single cavitation bubble in a wide range of parameter domain. Through analyzing the results more comprehensive knowledge would be available about its complex and nonlinear dynamics. 2. The bubble model The bubble model used for the numerical simulation was derived in [16] which is a modified model of Keller–Miksis equation [6] formulated by Prosperetti [7] and is given by (1):
1
! ! _ R_ € þ 3 R_ 2 1 R ¼ P þ 1 d ðRPÞ ¼ RR 2 q qc dt c 3c
1þ
! R dP R_ P þ c q qc dt
ð1Þ
with _ tÞ ¼ PðR; R;
P stat Pm þ
2r R0
3k R0 2r R_ 4l P stat þ Pm P a sinð2pmtÞ; R R R
ð2Þ
where R0 ¼ 10 lm is the equilibrium radius of the bubble, m is the frequency of the driving sound field, Pa is the amplitude of N is the the driving pressure, Pstat ¼ 100 KPa is the static ambient pressure, P m ¼ 2:33 KPa is the vapor pressure, r ¼ 0:0725 m kg Ns m surface tension, q ¼ 998 m3 is the liquid density, l ¼ 0:000001 m3 is the viscosity, c ¼ 1500 s is the sound velocity, and k ¼ 43 is the polytropic exponent of the gas in the bubble. These values were used for the numerical simulations unless otherwise mentioned. Between control parameters m and P a are the most important ones and the correspondent values are stated. 3. Stability analysis For stability analysis, it is convenient to transform the second-order differential equation (1) into an autonomous system of first-order differential equation of the following form: x_ ¼ y; 2 y y y P stat P m 2r x0 3k 2r y 3 þ þ 1 þ ð1 3kÞ 4l y_ ¼ 2 q x c c qx0 qx qx 1 y P stat Pm þ P a sinð2phÞ 2pmP a y 4l cosð2phÞ 1 xþ ; x 1 qc c q c qc h_ ¼ m;
ð3Þ
or equivalently dV ¼ FðV; aÞ; dt
ð4Þ
where h is the cyclic variable, Vðx; y; hÞ an autonomous vector field and aðR0 ; P a ; P stat ; P m ; m; l; q; c; kÞ is an element of the parameter space. This system generates a flow U ¼ fUT g on the phase space M ¼ R2 S and there exists a global map: X X P: ! ; c
c
V P ðx; y; hÞ ! PðV P Þ ¼ fUT gjPðx;y;h0 Þ c
1 , m
with T ¼ h0 is a constant determining the Poincare cross-section and ðx; yÞ the coordinates of the attractors in the Poincare P cross-section c , which is defined by X ¼ fðx; y; hÞ 2 R2 S1 j h ¼ h0 g: c
The choice of Poincare section is arbitrary; the only necessary condition is that the trajectory should cross the section P once every acoustic cycle. For driven oscillators like the bubble model, a natural way to define is to cut the torus like state space M transversally to the cyclic h direction at a fixed value h0 of h [16]. In this paper, the stability of a single cavitation bubble is studied versus the driving pressure amplitude, driving frequency, initial radius of the bubble, its surface tension and viscosity of the media.
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3.1. Lyapunov exponent spectrum The Lyapunov exponent is a quantitative measure of chaotic dynamics of a system by examining its very sensitive dependence on initial conditions. The Lyapunov exponent of a trajectory measures the average rate of divergence of neighboring trajectories. The Lyapunov exponent is defined as follows: Consider two nearest neighboring points in phase space at time 0 and t, with distances of the points in the ith direction kdxi ð0Þk and kdxi ðtÞk, respectively. The Lyapunov exponent is then defined by the average growth rate ki of the initial distance, ki ¼ lim
t!1
1 kdxi ðtÞk : ln t kdxi ð0Þk
ð5Þ
The existence of a positive lyapunov exponent is the indicator of chaos showing neighboring points with infinitesimal differences at the initial state abruptly separate from each other in the ith direction. Using the algorithm of Wolf et al. [33] and applying the Runge–Kutta method for solving (3), the Lyapunov exponent was calculated versus a given control parameter. Then the value of the control parameter increased a little and the Lyapunov exponent was calculated for the new control parameter. By continuing this procedure Lyapunov exponent spectrum of the system was plotted versus the control parameter.
3.2. Bifurcation diagrams Bifurcation means a qualitative change in the dynamical behavior of a system when a parameter of the system is varied. A bifurcation diagram provides a useful insight into the transition between different types of motion that can occur as one parameter of the system alters. It enables one to study the behavior of the system on a wide range of an interested control parameter. In this paper the dynamical behavior of the system is studied through plotting the bifurcation diagrams of the normalized radius of the bubble versus different control parameters. The procedure is as the following: The autonomous system of differential equation (3) was solved numerically applying the Runge–Kutta method. After the system reached its steady state then up to 100 orbits of xx0 in the h0 ¼ 0 of the Poincare plane were plotted in the bifurcation diagram versus the control parameter. This procedure continued by increasing the control parameter and the new resulting points were plotted in the bifurcation diagram versus the new control parameter. The process through analyzing the nonlinear transitions according to bifurcation diagrams of the system is divided into five categories: a. effect of pressure The normalized radius of a bubble with initial radius of 10 lm was plotted versus the applied pressure as the control parameter in the range of 10 KPa–2 MPa for different values of frequency. b. effect of frequency With the same initial radius as in a the bifurcation diagrams of the normalized radius of the bubble were plotted against the frequency as the control parameter. The range of the frequency studied here is from 10 KHz–2 MHz for different values of applied pressure. c. effect of initial radius For a fixed value of the frequency ranging between 0.1–1 MHz and 0.1–1 MPa, respectively the normalized bubble radius was sketched versus its initial radius. The range of the initial radius values considered here is from 0.5 to 50 lm. d. effect of surface tension For a fixed value of the frequency and pressure of 500 KHz and 500 KPa, respectively, but this time surface tension as the control parameter the bifurcation diagram of the normalized radius of the bubble was plotted. The range of the surface tension considered here is 0.01–8 N/m. e. effect of viscosity of the fluid Using the same initial conditions as in d, but taking the viscosity as the control parameter, the bifurcation diagram of the normalized bubble radius was plotted. The range of the viscosity considered here is from 1 l to 0.15 N s/m3.
4. Results and discussion 1. Lyapunov exponent spectra: Fig. 1a and b shows the bifurcation diagram and the Lyapunov spectrum of the bubble versus pressure while the driving frequency is 500 KHz. The range of the studied pressure is from 10 KPa to 2 MPa. According to the pictures the motion starts with period one up to 188 KPa, ensuing a period doubling till 225 KPa, where the motion undergoes a saddle node bifurcation pursuing four and then eight coexisting attractor state. At 270 KPa oscillations become chaotic. After the first chaotic window the system exhibits a period three oscillation at 524 KPa, which is followed by period six and 12 before the second transition to chaos. The system becomes extremely chaotic which lasts the whole remaining pressure interval between 740 KPa and 2 MPa with a window of complex periodic behavior. The very narrow window indicates the extreme sensitivity to initial applied pressure at these parameter values, therefor a small change in the magnitude
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Fig. 1. Bifurcation diagrams and the corresponding Lyapuonov spectrum of a bubble with 10 lm initial radius versus pressure and frequency, (a) control parameter is pressure (10 KPa—2 MPa) while the driving frequency is 500 KHz, (b) corresponding Lypunov spectrum of (a) and (c) control parameter is frequency (100 KHz—2 MHz) while the driving pressure is 500 KPa, (d) corresponding Lyapunov spectrum of (c).
of the driving pressure transforms the stable oscillations of the bubble to chaotic ones. As it is clearly seen in Fig. 1b, the corresponding Lyapunov exponent spectrum gives a quantitative measure on the behavior of the system with positive, negative and zero values evincing the chaotic, stable and quasi-periodic motion, respectively. Fig. 1c and d presents the bifurcation diagram and Lyapunov spectrum of the bubble versus frequency. The range of the studied frequency is from 100 KHz to 2 MHz. They show intermittent chaotic and stable behavior by frequency increase and the transformations from stable to chaotic oscillations are through period doubling bifurcations. The oscillations get completely stable after a threshold frequency through an inverse period doubling. Also this is confirmed by its Lyapunov spectrum. 2. Effect of pressure variations: Fig. 2a–d shows the bifurcation diagrams of the bubble versus pressure while the frequency of the sound field is 0:1; 0:2; 0:3; 0:6; 1 and 2 MHz, respectively. Stable and chaotic oscillations are presented in all of them along with simple period 1 to period 10 behaviors and transitions between different types of motions via period doubling, saddle node bifurcations to coexisting attractors and inverse period doubling (see Fig. 3). There are also windows of complex behavior with periods 2–8 oscillations in Fig. 2a–c. After the first transition to chaos the bubble begins its stable oscillation with period one, two, three and four before its motion becomes chaotic for the second time. As it is seen, bubbles with higher frequencies are stable in higher driving pressures. Regarding Fig. 2a–f, amplifying frequency implicates different transitional behavior compared to Fig. 2a–d. This property manifests itself specially in Fig. 2f, that the primary stable range of pressure is the widest. For 1 MHz of frequency the bubble is stable of period one first, but increasing the pressure causes its conversion to a coexisting attractor through a saddle node bifurcation exhibiting a period two behavior for a small interval. Then the bubble resumes its period one oscillation with a larger amplitude. A sequence of saddle node bifurcations happen again with a jump toward a coexisting attractor with smaller amplitude. These attractors coexist while the larger one encounter a period doubling and after a while the motion gets chaotic. Four coexisting attractors appear after a narrow window of chaotic behavior. Every attractor at this phase undergoes period doubling and the motion gets chaotic again. For 2 MHz of frequency the bubble stays stable with period one in a wide range of driving pressure from 10 KPa to 1.4 MPa. At 1.4 MPa through saddle node bifurcations 10 coexisting attractors appear and the system continues with period 11, and a slight period doubling arises after which one can witness the chaotic behavior. For clarifying the complex bubble behavior we have magnified the bifurcation diagram of a bubble with 4.5 lm initial radius driven with 200 KHz of frequency in Fig. 3 from 0.8 to 140 KPa. It is observed that different types of motion exist including period three, six, seven, thirteen, chaotic and transitions through period doubling and reverse period doubling.
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Fig. 2. Bifurcation diagrams of normalized bubble radius with initial radius of 10 lm versus pressure while the frequency is (a) 100 KHz, (b) 200 KHz, (c) 300 KHz, (d) 600 KHz, (e) 1 MHz and (f) 2 MHz.
3. Effect of frequency variations: Fig. 4a–f shows the effect of frequency variations on the dynamical behavior of a bubble with 10 lm initial radius when the applied pressure amplitude is 0:05; 0:1; 0:3; 0:5; 1 and 2 MPa, respectively. The range of the studied frequency was from 10 KHz to 2 MHz. However, the figures presented here are in the range of 100 KHz–2 MHz, because the large amplitude of oscillations in frequencies lower than 100 KHz decreases the visual sight for the detailed analysis of the nonlinear transitions. Results reveal that for small values of pressure the motion of the bubble is almost stable (Fig. 4a and b). The transitions between attractors are just through saddle node bifurcations. The amplitude of oscillations decreases rapidly with frequency increase, then the motion jumps to an attractor with higher amplitude and again the amplitude descends suddenly and bubble advances with a period one motion for good. For higher pressures chaotic behavior emerges in bubble response. In Fig. 4d–f one can observe intermittent chaotic and stable behavior. The transitions from stable to chaotic oscillations are via period doubling.
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Fig. 3. Magnification of the dynamics of a bubble with 4.5 lm initial radius driven with 200 KHz of frequency.
In accordance with Fig. 4c–d, after intermittent chaotic windows the bubble first begins period two oscillations which is followed by period doubling to period four and then chaos. By increasing the pressure the stable behavior between intermittent chaotic windows becomes of period three thereafter under period doubling turns into period six preceding chaos. At higher frequencies all of them become stable exhibiting a period one behavior. However, bubbles driven with higher pressures get stable at higher frequencies. The interesting point is that the last transition to stable behavior is via inverse period doubling followed by a saddle node bifurcation. In line with Fig. 4a–f, frequency growth initiates oscillation abatement. These figures highlight the stabilizing effect of higher frequencies. The regions of stable behavior between chaotic transitions become narrower in higher pressures, indicating more sensitivity to frequency changes. 4. Initial bubble radius: Fig. 5a–f shows the bifurcation diagrams of the normalized bubble radius when initial radius of the bubble is taken as the control parameter. The left column (Fig. 5a–c) illustrates the behavior of the bubble in a sound field with the amplitude 500 KPa and frequency of 100 KHz, 500 KHz and 1 MHz, respectively. Results reveal that for low values of frequency and small initial radii the bubble is stable. By increasing the bubble initial radius, a period doubling sequence is followed with a transition to chaos. At higher frequencies the chaotic region is small and as the bubble initial radius increases bubble reaches to a complete period one stability. In conformity with Fig. 5a–f the bubble driven with 1 MHz of frequency reaches stability in lower initial radii compared to the other two cases. The amplitude of oscillations is large with low frequencies when driven by the same pressure and initial radius. The figures introduce intermittent chaotic and stable behaviors. Between intermittent chaotic windows the bubble first begins with period two, then after pursuing period four through period doubling passes into chaos. The transition to the enduring stable behavior is through inverse period doubling. Intermittent occurrence of coexisting attractors is also observed through saddle node bifurcation. The right column (Fig. 5d–f) demonstrates the behavior of the bubble in a sound field when driven with the frequency of 300 KHz and pressures of 100 KPa, 300 KPa and 1 MPa, respectively. Results reveal that increasing the bubble initial radius decreases the amplitude of normalized oscillations as also seen in (Fig. 5a–c). According to the figures in low pressures chaotic windows are fewer than the ones in high pressures and the oscillations are stable associated with saddle node bifurcations rather than chaotic transitions. Pressure increase entails more intermittent chaotic windows and less stable state. Also pressure increase causes larger oscillation amplitudes. The interesting phenomenon to mention is the coincidence of period doubling and inverse period doubling processes in Fig. 5e. As we observe when we take initial radii as the control parameter the last transitions to stability from chaos are through inverse period doubling. In the end, considering Fig. 5a–f, reveals the stabilizing role of frequency and initial bubble radii increase, whereas increasing the pressure has the reverse implementation. Comparing Figs. 5a–f and 4a–f indicates a similarity in the bifurcation diagrams of the normalized radius of the bubble versus the driving frequency and initial bubble radius. This suggests similar dynamical behavior of the bubble accounting frequency and initial bubble radius changes. Such a phenomenon is also observed experimentally in [14], where the dynamics occurred spontaneously, i.e., while the external controllable parameters were held constant. It is stated in [14], that the most likely explanation is that the changes of R0 is dynamically the same as varying the driving frequency with fixed R0 . This phenomenon is also numerically confirmed here (see Figs. 4a–f and 5a–f). 5. Effect of surface tension: Fig. 6a–f display the bifurcation diagrams of the normalized bubble radius versus surface tension in the range of 0.1–8 N/m. The applied frequency is 500 KHz with the pressure of 500 KPa. The initial radii of the bubble are 1; 2; 5; 6; 8 and 10 lm. Results show that the motion of the bubble can be in specific ranges stable and chaotic. The transitions to strange attractors from stable fixed points are via the inverse period doubling and saddle node bifurcations.
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Fig. 4. Bifurcation diagrams of normalized bubble radius with initial radius of 10 lm versus frequency while the pressure is (a) 50 KPa, (b) 100 KPa, (c) 300 KPa, (d) 800 KPa, (e) 1 MPa and (f) 1:5 MPa.
Comparing Figs. 2a–f and 5a–f reveal that the outcoming bifurcation development of increasing surface tension is in the reverse direction of pressure increase. For bubbles with small initial radii the stabilizing role of surface tension is significant. Lower values of surface tension are needed to stabilize small bubbles rather than the bigger ones. As is seen for the same value of surface tension small bubbles exhibit extreme stability in comparison with big ones. These results strongly confirm the observation of the strong influence of surface tension on the dynamics of small bubbles presented in [34]. For lower values of surface tension the amplitude of oscillations are much larger for small bubbles, however, following a little increase in surface tension, the amplitude of oscillations abate suddenly indicating the strong influence of surface tension on them. 6. Viscosity effects: Fig. 7a–f represents the bifurcation diagrams of the normalized bubble radius versus viscosity when the bubble is driven with the frequency and pressure of 500 KHz and 500 KPa. They imply that for larger bubbles the transition from chaos to stable periodic oscillations are via the inverse period doubling. Small bubbles (<2 lm) are influenced more potently with viscosity increase compared with larger ones. It shows viscous effects dominate for small bubbles as also
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Fig. 5. Bifurcation diagrams of normalized bubble radius versus its initial radius with the frequency and pressure of: (a) 100 KHz; 500 KPa, (b) 500 KHz; 500 KPa, (c) 1 MHz; 500 KPa, (d) 300 KHz; 100 KPa, (e) 300 KHz; 300 KPa, (f) 300 KHz; 1 MPa.
mentioned in [3,8]. Similar to surface tension, the effect of viscosity increase is in the opposite direction of pressure increase. An interesting phenomenon is observed in Fig. 7f; after chaotic feature, the bubble undergoes six inverse period doubling and exhibits period six oscillations. Two of these inverse period doubled courses, jump with two saddle node bifurcations to two other inverse period doubling. At last the oscillations become of period four for good. 5. Implosion and shock wave phenomena After the generation of the cavitation bubbles in the liquid they begin their nonlinear oscillations. They enlarge and violently collapse, which results implosions that cause shock waves to be radiated from the sites of the collapse. During the inertial collapse the motion of the bubble wall may become supersonic and generate a spherically shock wave in the medium surrounding the collapsing bubble. The implosion causes the gases and vapors inside the bubbles com-
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pressed, generating intense heat (thousands of degrees Kelvin) and pressures (hundreds of atmospheres), that raises the temperature of the liquid surrounding the bubble immediately, and creates a local hot spot, spatially near the collapsing bubble and is limited temporally to the duration of the collapse. The production of high temperatures and pressures during inertial collapse can lead to formation of free radicals by ionizing the gas content. This has applications to sonochemistry by providing a suitable environment for chemical reactions. There also exists another type of shock waves capable to ionize the gas content called magnetohydrodynamic shock waves. The existence of such structure is also proved for this phenomenon [35] as well as acoustic cavitation [17] and implosions [36]. The propagation of shock waves resulting from a strong explosion and the associated instantaneous release of energy has been discussed in detail in [37]. Another interesting phenomenon associated with collapsing bubbles is the evidence for nuclear reactions in imploding bubbles [38]. This is called sonofusion [25]. Tiny bubbles imploded by sound waves can make hydrogen nuclei fuse and may one day become a useful and safe new energy source for mankind [25]. The implosion creates spherical shock waves
Fig. 6. Bifurcation diagrams of normalized bubble radius versus surface tension with 500 KHz of frequency and 500 KPa of pressure with initial radius of: (a) 1 lm, (b) 2 lm, (c) 5 lm, (d) 6 lm, (e) 8 lm, (f) 10 lm.
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Fig. 7. Bifurcation diagrams of normalized bubble radius versus viscosity initial when the driving frequency and pressure are, respectively, 500 KHz and 500 KPa for bubbles with initial radii of: (a) 1 lm, (b) 2 lm, (c) 5 lm, (d) 6 lm, (e) 8 lm, (f) 10 lm.
within the bubbles that travel inward at high speeds and significantly strengthen as they converge to center. In the presence of a thermonuclear material, implosion can provide high densities and temperatures needed for certain nuclear reactions to occur. There is also a great deal of contributions to nuclear reactions during implosions published by El Naschie in recent years [36,39–41]. 6. Conclusion In this article through applying the methods of chaos physics, the nonlinear behavior of an acoustically driven spherical bubble was studied. Our results show its rich nonlinear dynamics with respect to variations in the control parameters of the system. The obtained results give confirmation to some of the phenomena detected in previous works and give a helpful insight into understanding the complex dynamics of the bubble.
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Another important phenomenon which needs to be investigated associating the bubble dynamics, is the behavior of a bubble cluster and the internal bubble-bubble interactions. This can be the subject of future studies. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41]
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Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
Nonlinear transitions of a spherical cavitation bubble Sohrab Behnia a,*, Amin Jafari b, Wiria Soltanpoor b, Okhtay Jahanbakhsh b a b
Department of Physics, IAU, Urmia, Iran Department of Physics, Urmia University, Urmia, Iran
a r t i c l e
i n f o
Article history: Accepted 31 March 2008
a b s t r a c t Dynamics of acoustically driven bubbles are known to be complex and uncontrollable. Depending on the applied pressure, frequency and the properties of the bubble and the host media, the radial oscillations of the bubble has been reported to be stable and chaotic. In this paper, the dynamics of a single acoustically driven bubble is widely studied applying the methods of chaos physics. The stability of the bubble is analyzed through plotting the bifurcation diagrams and Lyapunov exponent spectra versus pressure, frequency, initial bubble radius, surface tension and viscosity as the control parameters. Results show the rich nonlinear dynamics of the bubble including period doubling, inverse period doubling, saddle node bifurcations, quasi-periodicity and chaos. Also similarities were detected in the bubble response to variations between some of the control parameters. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction A gas bubble driven in motion by ultrasound is an example of a system with highly nonlinear properties in which deterministic chaos manifests itself. This phenomenon occurs when a high amplitude, high frequency sound is applied to the liquid [1–3]. While high intensity ultrasound is introduced to an aqueous solution, the rare faction cycle may exceed the attractive forces among liquid molecules making cavitation bubbles occur and begin their nonlinear oscillations which is accompanied by intense noise emission; the acoustic cavitation noise [4]. The noise is found not to be caused by statistical rupture process in the liquid but to be of deterministic origin [5] belonging to the class of deterministically chaotic systems [4]. From theoretical point of view, this phenomenon even in its simplest forms is described by a highly nonlinear differential equation of second order [6–9]. Experimentally Esche [10] was the first one to observe subharmonics in the spectrum of the response of liquid in 1952. More after Lauterborn and Cramer [5] observed the subharmonics route to chaos including period doubling bifurcations up to f =8 and reverse bifurcations in the acoustic cavitation noise. Later the bifurcation structure of cavitation was studied numerically and a superstructure resulting from nonlinear resonance was found with period doubling Feigenbaum direct cascades [11] and Grossmann inverse cascades [12] as fine structure. Holographic observations of period doubling and chaotic bubble oscillations in [13] confirmed the view that nonlinear acoustic response of the liquid is mediated by the bubbles and their nonlinear oscillations. The phenomenon of chaotic bubble oscillations was also observed by measuring the time delays between flashes of emitted light in an experiment on sonoluminescence showing period doubling, quasiperiodicity and chaos [14]. The rich nonlinear dynamics of the system was also numerically investigated through applying the methods of chaos physics [15–19]. Besides its complex and nonlinear behavior, acoustic cavitation gas bubbles are present inevitably in lots of applications including material science, sonoluminescence and sonochemistry [20–24], sonofusion [25] and medical procedures such as: lithotripsy, diagnostic imaging, drug and gene delivery, increasing membrane permeability, opening blood brain barrier and
* Corresponding author. E-mail address: [email protected] (S. Behnia). 0960-0779/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2008.04.011
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high intensity focused ultrasound surgery [26–30]. A complete understanding of the dynamics of the phenomena is deemed necessary in order to achieve their full potential use in industry and medicine. For this purpose it is first needed to expand our knowledge on a single bubble behavior under different possible conditions including the frequency and pressure variations of the sound field, fluid and gas properties such as viscosity and surface tension and bubble properties like its initial radius. The results can help us choose a regime to avoid chaotic motion because when a system falls into chaos its behavior is difficult to predict and control [31,32]. In this paper the effects of parameters mentioned above are studied for a single cavitation bubble in a wide range of parameter domain. Through analyzing the results more comprehensive knowledge would be available about its complex and nonlinear dynamics. 2. The bubble model The bubble model used for the numerical simulation was derived in [16] which is a modified model of Keller–Miksis equation [6] formulated by Prosperetti [7] and is given by (1):
1
! ! _ R_ € þ 3 R_ 2 1 R ¼ P þ 1 d ðRPÞ ¼ RR 2 q qc dt c 3c
1þ
! R dP R_ P þ c q qc dt
ð1Þ
with _ tÞ ¼ PðR; R;
P stat Pm þ
2r R0
3k R0 2r R_ 4l P stat þ Pm P a sinð2pmtÞ; R R R
ð2Þ
where R0 ¼ 10 lm is the equilibrium radius of the bubble, m is the frequency of the driving sound field, Pa is the amplitude of N is the the driving pressure, Pstat ¼ 100 KPa is the static ambient pressure, P m ¼ 2:33 KPa is the vapor pressure, r ¼ 0:0725 m kg Ns m surface tension, q ¼ 998 m3 is the liquid density, l ¼ 0:000001 m3 is the viscosity, c ¼ 1500 s is the sound velocity, and k ¼ 43 is the polytropic exponent of the gas in the bubble. These values were used for the numerical simulations unless otherwise mentioned. Between control parameters m and P a are the most important ones and the correspondent values are stated. 3. Stability analysis For stability analysis, it is convenient to transform the second-order differential equation (1) into an autonomous system of first-order differential equation of the following form: x_ ¼ y; 2 y y y P stat P m 2r x0 3k 2r y 3 þ þ 1 þ ð1 3kÞ 4l y_ ¼ 2 q x c c qx0 qx qx 1 y P stat Pm þ P a sinð2phÞ 2pmP a y 4l cosð2phÞ 1 xþ ; x 1 qc c q c qc h_ ¼ m;
ð3Þ
or equivalently dV ¼ FðV; aÞ; dt
ð4Þ
where h is the cyclic variable, Vðx; y; hÞ an autonomous vector field and aðR0 ; P a ; P stat ; P m ; m; l; q; c; kÞ is an element of the parameter space. This system generates a flow U ¼ fUT g on the phase space M ¼ R2 S and there exists a global map: X X P: ! ; c
c
V P ðx; y; hÞ ! PðV P Þ ¼ fUT gjPðx;y;h0 Þ c
1 , m
with T ¼ h0 is a constant determining the Poincare cross-section and ðx; yÞ the coordinates of the attractors in the Poincare P cross-section c , which is defined by X ¼ fðx; y; hÞ 2 R2 S1 j h ¼ h0 g: c
The choice of Poincare section is arbitrary; the only necessary condition is that the trajectory should cross the section P once every acoustic cycle. For driven oscillators like the bubble model, a natural way to define is to cut the torus like state space M transversally to the cyclic h direction at a fixed value h0 of h [16]. In this paper, the stability of a single cavitation bubble is studied versus the driving pressure amplitude, driving frequency, initial radius of the bubble, its surface tension and viscosity of the media.
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3.1. Lyapunov exponent spectrum The Lyapunov exponent is a quantitative measure of chaotic dynamics of a system by examining its very sensitive dependence on initial conditions. The Lyapunov exponent of a trajectory measures the average rate of divergence of neighboring trajectories. The Lyapunov exponent is defined as follows: Consider two nearest neighboring points in phase space at time 0 and t, with distances of the points in the ith direction kdxi ð0Þk and kdxi ðtÞk, respectively. The Lyapunov exponent is then defined by the average growth rate ki of the initial distance, ki ¼ lim
t!1
1 kdxi ðtÞk : ln t kdxi ð0Þk
ð5Þ
The existence of a positive lyapunov exponent is the indicator of chaos showing neighboring points with infinitesimal differences at the initial state abruptly separate from each other in the ith direction. Using the algorithm of Wolf et al. [33] and applying the Runge–Kutta method for solving (3), the Lyapunov exponent was calculated versus a given control parameter. Then the value of the control parameter increased a little and the Lyapunov exponent was calculated for the new control parameter. By continuing this procedure Lyapunov exponent spectrum of the system was plotted versus the control parameter.
3.2. Bifurcation diagrams Bifurcation means a qualitative change in the dynamical behavior of a system when a parameter of the system is varied. A bifurcation diagram provides a useful insight into the transition between different types of motion that can occur as one parameter of the system alters. It enables one to study the behavior of the system on a wide range of an interested control parameter. In this paper the dynamical behavior of the system is studied through plotting the bifurcation diagrams of the normalized radius of the bubble versus different control parameters. The procedure is as the following: The autonomous system of differential equation (3) was solved numerically applying the Runge–Kutta method. After the system reached its steady state then up to 100 orbits of xx0 in the h0 ¼ 0 of the Poincare plane were plotted in the bifurcation diagram versus the control parameter. This procedure continued by increasing the control parameter and the new resulting points were plotted in the bifurcation diagram versus the new control parameter. The process through analyzing the nonlinear transitions according to bifurcation diagrams of the system is divided into five categories: a. effect of pressure The normalized radius of a bubble with initial radius of 10 lm was plotted versus the applied pressure as the control parameter in the range of 10 KPa–2 MPa for different values of frequency. b. effect of frequency With the same initial radius as in a the bifurcation diagrams of the normalized radius of the bubble were plotted against the frequency as the control parameter. The range of the frequency studied here is from 10 KHz–2 MHz for different values of applied pressure. c. effect of initial radius For a fixed value of the frequency ranging between 0.1–1 MHz and 0.1–1 MPa, respectively the normalized bubble radius was sketched versus its initial radius. The range of the initial radius values considered here is from 0.5 to 50 lm. d. effect of surface tension For a fixed value of the frequency and pressure of 500 KHz and 500 KPa, respectively, but this time surface tension as the control parameter the bifurcation diagram of the normalized radius of the bubble was plotted. The range of the surface tension considered here is 0.01–8 N/m. e. effect of viscosity of the fluid Using the same initial conditions as in d, but taking the viscosity as the control parameter, the bifurcation diagram of the normalized bubble radius was plotted. The range of the viscosity considered here is from 1 l to 0.15 N s/m3.
4. Results and discussion 1. Lyapunov exponent spectra: Fig. 1a and b shows the bifurcation diagram and the Lyapunov spectrum of the bubble versus pressure while the driving frequency is 500 KHz. The range of the studied pressure is from 10 KPa to 2 MPa. According to the pictures the motion starts with period one up to 188 KPa, ensuing a period doubling till 225 KPa, where the motion undergoes a saddle node bifurcation pursuing four and then eight coexisting attractor state. At 270 KPa oscillations become chaotic. After the first chaotic window the system exhibits a period three oscillation at 524 KPa, which is followed by period six and 12 before the second transition to chaos. The system becomes extremely chaotic which lasts the whole remaining pressure interval between 740 KPa and 2 MPa with a window of complex periodic behavior. The very narrow window indicates the extreme sensitivity to initial applied pressure at these parameter values, therefor a small change in the magnitude
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Fig. 1. Bifurcation diagrams and the corresponding Lyapuonov spectrum of a bubble with 10 lm initial radius versus pressure and frequency, (a) control parameter is pressure (10 KPa—2 MPa) while the driving frequency is 500 KHz, (b) corresponding Lypunov spectrum of (a) and (c) control parameter is frequency (100 KHz—2 MHz) while the driving pressure is 500 KPa, (d) corresponding Lyapunov spectrum of (c).
of the driving pressure transforms the stable oscillations of the bubble to chaotic ones. As it is clearly seen in Fig. 1b, the corresponding Lyapunov exponent spectrum gives a quantitative measure on the behavior of the system with positive, negative and zero values evincing the chaotic, stable and quasi-periodic motion, respectively. Fig. 1c and d presents the bifurcation diagram and Lyapunov spectrum of the bubble versus frequency. The range of the studied frequency is from 100 KHz to 2 MHz. They show intermittent chaotic and stable behavior by frequency increase and the transformations from stable to chaotic oscillations are through period doubling bifurcations. The oscillations get completely stable after a threshold frequency through an inverse period doubling. Also this is confirmed by its Lyapunov spectrum. 2. Effect of pressure variations: Fig. 2a–d shows the bifurcation diagrams of the bubble versus pressure while the frequency of the sound field is 0:1; 0:2; 0:3; 0:6; 1 and 2 MHz, respectively. Stable and chaotic oscillations are presented in all of them along with simple period 1 to period 10 behaviors and transitions between different types of motions via period doubling, saddle node bifurcations to coexisting attractors and inverse period doubling (see Fig. 3). There are also windows of complex behavior with periods 2–8 oscillations in Fig. 2a–c. After the first transition to chaos the bubble begins its stable oscillation with period one, two, three and four before its motion becomes chaotic for the second time. As it is seen, bubbles with higher frequencies are stable in higher driving pressures. Regarding Fig. 2a–f, amplifying frequency implicates different transitional behavior compared to Fig. 2a–d. This property manifests itself specially in Fig. 2f, that the primary stable range of pressure is the widest. For 1 MHz of frequency the bubble is stable of period one first, but increasing the pressure causes its conversion to a coexisting attractor through a saddle node bifurcation exhibiting a period two behavior for a small interval. Then the bubble resumes its period one oscillation with a larger amplitude. A sequence of saddle node bifurcations happen again with a jump toward a coexisting attractor with smaller amplitude. These attractors coexist while the larger one encounter a period doubling and after a while the motion gets chaotic. Four coexisting attractors appear after a narrow window of chaotic behavior. Every attractor at this phase undergoes period doubling and the motion gets chaotic again. For 2 MHz of frequency the bubble stays stable with period one in a wide range of driving pressure from 10 KPa to 1.4 MPa. At 1.4 MPa through saddle node bifurcations 10 coexisting attractors appear and the system continues with period 11, and a slight period doubling arises after which one can witness the chaotic behavior. For clarifying the complex bubble behavior we have magnified the bifurcation diagram of a bubble with 4.5 lm initial radius driven with 200 KHz of frequency in Fig. 3 from 0.8 to 140 KPa. It is observed that different types of motion exist including period three, six, seven, thirteen, chaotic and transitions through period doubling and reverse period doubling.
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Fig. 2. Bifurcation diagrams of normalized bubble radius with initial radius of 10 lm versus pressure while the frequency is (a) 100 KHz, (b) 200 KHz, (c) 300 KHz, (d) 600 KHz, (e) 1 MHz and (f) 2 MHz.
3. Effect of frequency variations: Fig. 4a–f shows the effect of frequency variations on the dynamical behavior of a bubble with 10 lm initial radius when the applied pressure amplitude is 0:05; 0:1; 0:3; 0:5; 1 and 2 MPa, respectively. The range of the studied frequency was from 10 KHz to 2 MHz. However, the figures presented here are in the range of 100 KHz–2 MHz, because the large amplitude of oscillations in frequencies lower than 100 KHz decreases the visual sight for the detailed analysis of the nonlinear transitions. Results reveal that for small values of pressure the motion of the bubble is almost stable (Fig. 4a and b). The transitions between attractors are just through saddle node bifurcations. The amplitude of oscillations decreases rapidly with frequency increase, then the motion jumps to an attractor with higher amplitude and again the amplitude descends suddenly and bubble advances with a period one motion for good. For higher pressures chaotic behavior emerges in bubble response. In Fig. 4d–f one can observe intermittent chaotic and stable behavior. The transitions from stable to chaotic oscillations are via period doubling.
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Fig. 3. Magnification of the dynamics of a bubble with 4.5 lm initial radius driven with 200 KHz of frequency.
In accordance with Fig. 4c–d, after intermittent chaotic windows the bubble first begins period two oscillations which is followed by period doubling to period four and then chaos. By increasing the pressure the stable behavior between intermittent chaotic windows becomes of period three thereafter under period doubling turns into period six preceding chaos. At higher frequencies all of them become stable exhibiting a period one behavior. However, bubbles driven with higher pressures get stable at higher frequencies. The interesting point is that the last transition to stable behavior is via inverse period doubling followed by a saddle node bifurcation. In line with Fig. 4a–f, frequency growth initiates oscillation abatement. These figures highlight the stabilizing effect of higher frequencies. The regions of stable behavior between chaotic transitions become narrower in higher pressures, indicating more sensitivity to frequency changes. 4. Initial bubble radius: Fig. 5a–f shows the bifurcation diagrams of the normalized bubble radius when initial radius of the bubble is taken as the control parameter. The left column (Fig. 5a–c) illustrates the behavior of the bubble in a sound field with the amplitude 500 KPa and frequency of 100 KHz, 500 KHz and 1 MHz, respectively. Results reveal that for low values of frequency and small initial radii the bubble is stable. By increasing the bubble initial radius, a period doubling sequence is followed with a transition to chaos. At higher frequencies the chaotic region is small and as the bubble initial radius increases bubble reaches to a complete period one stability. In conformity with Fig. 5a–f the bubble driven with 1 MHz of frequency reaches stability in lower initial radii compared to the other two cases. The amplitude of oscillations is large with low frequencies when driven by the same pressure and initial radius. The figures introduce intermittent chaotic and stable behaviors. Between intermittent chaotic windows the bubble first begins with period two, then after pursuing period four through period doubling passes into chaos. The transition to the enduring stable behavior is through inverse period doubling. Intermittent occurrence of coexisting attractors is also observed through saddle node bifurcation. The right column (Fig. 5d–f) demonstrates the behavior of the bubble in a sound field when driven with the frequency of 300 KHz and pressures of 100 KPa, 300 KPa and 1 MPa, respectively. Results reveal that increasing the bubble initial radius decreases the amplitude of normalized oscillations as also seen in (Fig. 5a–c). According to the figures in low pressures chaotic windows are fewer than the ones in high pressures and the oscillations are stable associated with saddle node bifurcations rather than chaotic transitions. Pressure increase entails more intermittent chaotic windows and less stable state. Also pressure increase causes larger oscillation amplitudes. The interesting phenomenon to mention is the coincidence of period doubling and inverse period doubling processes in Fig. 5e. As we observe when we take initial radii as the control parameter the last transitions to stability from chaos are through inverse period doubling. In the end, considering Fig. 5a–f, reveals the stabilizing role of frequency and initial bubble radii increase, whereas increasing the pressure has the reverse implementation. Comparing Figs. 5a–f and 4a–f indicates a similarity in the bifurcation diagrams of the normalized radius of the bubble versus the driving frequency and initial bubble radius. This suggests similar dynamical behavior of the bubble accounting frequency and initial bubble radius changes. Such a phenomenon is also observed experimentally in [14], where the dynamics occurred spontaneously, i.e., while the external controllable parameters were held constant. It is stated in [14], that the most likely explanation is that the changes of R0 is dynamically the same as varying the driving frequency with fixed R0 . This phenomenon is also numerically confirmed here (see Figs. 4a–f and 5a–f). 5. Effect of surface tension: Fig. 6a–f display the bifurcation diagrams of the normalized bubble radius versus surface tension in the range of 0.1–8 N/m. The applied frequency is 500 KHz with the pressure of 500 KPa. The initial radii of the bubble are 1; 2; 5; 6; 8 and 10 lm. Results show that the motion of the bubble can be in specific ranges stable and chaotic. The transitions to strange attractors from stable fixed points are via the inverse period doubling and saddle node bifurcations.
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Fig. 4. Bifurcation diagrams of normalized bubble radius with initial radius of 10 lm versus frequency while the pressure is (a) 50 KPa, (b) 100 KPa, (c) 300 KPa, (d) 800 KPa, (e) 1 MPa and (f) 1:5 MPa.
Comparing Figs. 2a–f and 5a–f reveal that the outcoming bifurcation development of increasing surface tension is in the reverse direction of pressure increase. For bubbles with small initial radii the stabilizing role of surface tension is significant. Lower values of surface tension are needed to stabilize small bubbles rather than the bigger ones. As is seen for the same value of surface tension small bubbles exhibit extreme stability in comparison with big ones. These results strongly confirm the observation of the strong influence of surface tension on the dynamics of small bubbles presented in [34]. For lower values of surface tension the amplitude of oscillations are much larger for small bubbles, however, following a little increase in surface tension, the amplitude of oscillations abate suddenly indicating the strong influence of surface tension on them. 6. Viscosity effects: Fig. 7a–f represents the bifurcation diagrams of the normalized bubble radius versus viscosity when the bubble is driven with the frequency and pressure of 500 KHz and 500 KPa. They imply that for larger bubbles the transition from chaos to stable periodic oscillations are via the inverse period doubling. Small bubbles (<2 lm) are influenced more potently with viscosity increase compared with larger ones. It shows viscous effects dominate for small bubbles as also
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Fig. 5. Bifurcation diagrams of normalized bubble radius versus its initial radius with the frequency and pressure of: (a) 100 KHz; 500 KPa, (b) 500 KHz; 500 KPa, (c) 1 MHz; 500 KPa, (d) 300 KHz; 100 KPa, (e) 300 KHz; 300 KPa, (f) 300 KHz; 1 MPa.
mentioned in [3,8]. Similar to surface tension, the effect of viscosity increase is in the opposite direction of pressure increase. An interesting phenomenon is observed in Fig. 7f; after chaotic feature, the bubble undergoes six inverse period doubling and exhibits period six oscillations. Two of these inverse period doubled courses, jump with two saddle node bifurcations to two other inverse period doubling. At last the oscillations become of period four for good. 5. Implosion and shock wave phenomena After the generation of the cavitation bubbles in the liquid they begin their nonlinear oscillations. They enlarge and violently collapse, which results implosions that cause shock waves to be radiated from the sites of the collapse. During the inertial collapse the motion of the bubble wall may become supersonic and generate a spherically shock wave in the medium surrounding the collapsing bubble. The implosion causes the gases and vapors inside the bubbles com-
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pressed, generating intense heat (thousands of degrees Kelvin) and pressures (hundreds of atmospheres), that raises the temperature of the liquid surrounding the bubble immediately, and creates a local hot spot, spatially near the collapsing bubble and is limited temporally to the duration of the collapse. The production of high temperatures and pressures during inertial collapse can lead to formation of free radicals by ionizing the gas content. This has applications to sonochemistry by providing a suitable environment for chemical reactions. There also exists another type of shock waves capable to ionize the gas content called magnetohydrodynamic shock waves. The existence of such structure is also proved for this phenomenon [35] as well as acoustic cavitation [17] and implosions [36]. The propagation of shock waves resulting from a strong explosion and the associated instantaneous release of energy has been discussed in detail in [37]. Another interesting phenomenon associated with collapsing bubbles is the evidence for nuclear reactions in imploding bubbles [38]. This is called sonofusion [25]. Tiny bubbles imploded by sound waves can make hydrogen nuclei fuse and may one day become a useful and safe new energy source for mankind [25]. The implosion creates spherical shock waves
Fig. 6. Bifurcation diagrams of normalized bubble radius versus surface tension with 500 KHz of frequency and 500 KPa of pressure with initial radius of: (a) 1 lm, (b) 2 lm, (c) 5 lm, (d) 6 lm, (e) 8 lm, (f) 10 lm.
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Fig. 7. Bifurcation diagrams of normalized bubble radius versus viscosity initial when the driving frequency and pressure are, respectively, 500 KHz and 500 KPa for bubbles with initial radii of: (a) 1 lm, (b) 2 lm, (c) 5 lm, (d) 6 lm, (e) 8 lm, (f) 10 lm.
within the bubbles that travel inward at high speeds and significantly strengthen as they converge to center. In the presence of a thermonuclear material, implosion can provide high densities and temperatures needed for certain nuclear reactions to occur. There is also a great deal of contributions to nuclear reactions during implosions published by El Naschie in recent years [36,39–41]. 6. Conclusion In this article through applying the methods of chaos physics, the nonlinear behavior of an acoustically driven spherical bubble was studied. Our results show its rich nonlinear dynamics with respect to variations in the control parameters of the system. The obtained results give confirmation to some of the phenomena detected in previous works and give a helpful insight into understanding the complex dynamics of the bubble.
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