Ultrasonics Sonochemistry 21 (2014) 1512–1518
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Hydrodynamic approach to multibubble sonoluminescence Shahid Mahmood, Yungpil Yoo, Jaekyoon Oh, Ho-Young Kwak ⇑ Mechanical Engineering Department, Chung-Ang University, Seoul 156-756, Republic of Korea
a r t i c l e
i n f o
Article history: Received 7 December 2013 Received in revised form 25 January 2014 Accepted 25 January 2014 Available online 2 February 2014 Keywords: Multibubble sonoluminescence Bubble cluster Radiation pressure Pulse width
a b s t r a c t The velocity profile and radiation pressure field of a bubble cluster containing several thousand micro bubbles were obtained by solving the continuity and momentum equations for the bubbly mixture. In this study, the bubbles in the cluster are assumed to be generated and collapsed synchronously with an applied ultrasound. Numerical calculations describing the behavior of a micro bubble in a cluster included the effect of the radiation pressure field from the synchronizing motion of bubbles in the cluster. The radiation pressure generated from surrounding bubbles affects the bubble’s behavior by increasing the effective mass of the bubble so that the bubble expands slowly to a smaller maximum size. The light pulse width and spectral radiance from a bubble in a cluster subjected to ultrasound were calculated by adding a radiation pressure term to the Keller–Miksis equation, and the values were compared to experimental values of the multibubble sonoluminescence condition. There was close agreement between the calculated and observed values. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction It is well known that in multi-bubble sonoluminescence (MBSL), several thousands of micro-bubbles are generated and collapsed synchronously with an applied ultrasound [1–3]. Recent measurements on the pulse width of a cloud of bubbles subjected to ultrasound using a time-correlated single photon counting technique indicated that the bubbles in a cloud collapse simultaneously to emit a light that is synchronous with the applied ultrasound [4]. A spherical bubble cloud subjected to harmonic far-field pressure excitation was investigated by Omta [5] and D’Agosta and Brennen [6]. They revealed that the natural frequency of the bubble cluster is always less than the natural frequency of the individual bubbles. Oguz and Prosperetti [7] investigated the interaction of two 100 lm bubbles subjected to an ultrasound with a moderate amplitude. Mettin et al. [8] considered the mutual interaction between two micro bubbles (R0 < 10 lm) in a strong acoustic field (Pa > 1 bar, fd = 20 kHz). They found that the strength and direction of the secondary Bjerknes forces due to the radiation generated by other bubbles differed from the forces predicted by linear theory. Yasui et al. [9] performed numerical simulations on a system of two bubbles and considered the interactions between n numbers of bubbles. They found that the expansion of a bubble during the rarefaction phase of ultrasound was strongly reduced by the presence of other bubbles in the cluster. They also obtained the ⇑ Corresponding author. Tel.: +82 28205278. E-mail address: [email protected] (H.-Y. Kwak). http://dx.doi.org/10.1016/j.ultsonch.2014.01.022 1350-4177/Ó 2014 Elsevier B.V. All rights reserved.
pressure field of the center of a cloud of similarly sized, homogeneously distributed bubbles which pulsated together with an applied ultrasound. An [10] obtained the radiation sound pressure from the other bubbles acting on a particular bubble in a cluster. He investigated the collective motion of similarly sized microbubbles in a cluster and found that the radiation pressure term added in the Keller–Miksis (KM) equation considerably suppresses bubble motion. Recently, Dzaharudin et al. [11] performed numerical simulations of a cluster of encapsulated microbubbles by adding the interaction term in the KM equation. They found that the oscillation amplitude of microbubbles that are close together was reduced for a given applied ultrasound power. In this study, MBSL is studied hydro-dynamically to obtain the velocity profile and radiation pressure field by solving the continuity and momentum equations for a spherical cluster containing numerous microbubbles. The calculated pulse width and spectral radiance for a bubble with the radiation pressure added in the KM equation are compared with the measured pulse width and spectral radiance values of the MBSL.
2. Single bubble behavior in an ultrasonic field 2.1. A set of analytical solutions for the Navier–Stokes equations The hydrodynamics related to the single bubble sonoluminescence phenomenon involves in solving the Navier–Stokes equations for the gas inside a bubble and the liquid adjacent to the
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bubble wall. The mass, momentum and energy equations for the gas inside a spherical bubble are given as;
@ qg 1 @ þ 2 q ug r2 ¼ 0; r @r g @t @P @ 1 @ ðqg ug Þ þ 2 qg u2g r2 þ b ¼ 0; @t r @r @r
qg C v ;b
DT b Pb d 2 1 d 2 r ug 2 ¼ 2 ðr qr Þ; r dr Dt r dr
ð1Þ
C¼ ð2Þ
ð3Þ
where r is the distance from center, qg is the gas density, ug is the gas velocity which obeys ug (Rb, t), the bubble wall velocity, Pb is the gas pressure, Cv,b is a constant volume specific heat and qr is the radial component of heat flux inside a bubble. A set of analytical solutions for the above conservation equations [12,13] is given as;
qg ¼ q0 þ qr ug ¼
R_ b r; Rb
Pb ¼ Pb0
ð4Þ ð5Þ
€ R 1 1 q0 þ qr b r 2 ; 2 2 Rb
TðrÞ ¼ T b ðrÞ þ T 0b ðrÞ
ð6Þ ð7Þ
where q0 is the gas density at bubble center and qr is the radial dependent gas density, which are given as q0 R3b ¼ const: and
qr ¼ ar2 =R5b ;respectively. The constant a is related to the gas mass inside a bubble and was taken as 5q0/(4p). Pb0 is the gas pressure at bubble center. The linear velocity profile describing the spatial in-homogeneities inside the bubble is a crucial ansatz for the homologous motion of a spherical object, which is encountered in another energy focusing mechanism of gravitational collapse [14]. The quadratic pressure profile given in Eq. (6), was recently verified by comparisons with direct numerical simulations [15]. The temperature profile due to the uniform pressure distribution Tb(r) is well known and is valid for a non-sonoluminescing gas bubble [16]:
2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 2 B 4 A A r 5 1 þ T b0 2g ðT bl T 1 Þ ; T b ðrÞ ¼ 1 þ A B B Rb
ð8Þ
where A and B are the are the coefficients of the temperaturedependent gas conductivity which has the form kg = AT + B [17], where g = (Rb/d)/(kl/B) and kl is the thermal conductivity of the liquid, d is thermal boundary layer thickness. The temperature at the bubble wall, Tbl in Eq. (9), can be obtained from Eq. (8) with a boundary condition of Tb Tb (Rb,t) = Tbl. The temperature is given as
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # B A A 2 2 T bl ¼ ð1 þ gÞ þ ð1 þ gÞ þ 2 T bo þ T þ gT 1 : A B 2B bo
The coefficient C(t) may be determined from the boundary con0 0 dition at the wall, kg dT b =dr ¼ kl dT l =dr;where Tl is the quadratic temperature distribution in the thermal boundary layer with a thickness d0 . That is,
h i ... 1 € b Rb þ Rb R2 ð3c 2ÞR_ b R b 20ðc 1Þ " # d0 5 Rb 5 q0 þ qr¼Rb þ 0 q0 þ qr¼Rb þ T 1 : 14 21 kl 2kg
The temperature distribution inside a bubble may be estimated 0 from Eqs. (7), (8), and (10) with appropriated values of d0 and kg , 0 where kg is gas conductivity in a dense plasma state [19]. The boundary layer thickness d0 may be determined from the relation 4pkl R2b ðT bl T 1 Þ=d0 ¼ P, where P is the power loss due to the brake radiation (bremsstrahlung) [12]. The temperature profile given in Eq. (10) yields a thermal spike when the acceleration of the bubble wall exceeds 1012 m/s2 [12] while the temperature distribution given in Eq. (8) provides a background temperature. The mass and momentum equation for the liquid adjacent to the bubble wall provides the well-known KM equation describing the motion for the bubble wall [20], which is valid when the bubble wall velocity does not exceed the speed of sound in the liquid. That is,
U b dU b 3 2 Ub Rb 1 þ Ub 1 2 C B dt 3C B
1 U b Rb d Rb PB Ps t þ P1 1þ þ ¼ q1 C B C B dt CB
ð12Þ
where Rb is the bubble radius, Ub is the bubble wall velocity, CB is the speed of sound in the liquid at the bubble wall, and q1 and P1 are the medium density and pressure, respectively. The liquid pressure on the external side of the bubble wall PB is related to the pressure inside the bubble wall Pb by PB = Pb 2r/Rb 4lUb/ Rb where r and l are the surface tension and dynamic viscosity of the liquid, respectively. The pressure of the driving sound field Ps may be represented by a sinusoidal function such as Ps = PAsin(xt) where PA is the driving sound amplitude, x = 2pfd and fd is frequency. For incompressible limit or Ub/CB ? 0, the KM equation reduces to the Rayleigh–Plesset equation. The mass and energy equation for the liquid provides a timedependent first order equation for the thermal boundary layer thickness d assuming a quadratic in temperature profile, which is given by [21],
"
" 2 # 2 # d 3 d dd 6a d 1 d dRb 1þ þ ¼ 2 þ Rb 10 Rb dt Rb 2 Rb d dt " 2 # 1 d 1 d 1 þ d 1þ 2 Rb 10 Rb T bl T 1
"
ð11Þ
dT bl dt
ð9Þ
ð13Þ
The temperature distribution given in Eq. (8) is valid until the characteristic time for the vibrational motion of the molecules is much less than the relaxation time for the translational motion of molecules [18]. The temperature rise and subsequent quenching due to bubble wall acceleration is given by:
where a is the thermal diffusivity of liquid. The above equation determines the heat flow rate through the bubble wall. The values for instantaneous bubble radius, bubble wall velocity and acceleration and the thermal boundary layer thickness obtained from Eqs. (12) and (13) are used to calculate the density with Eq. (4), velocity with Eq. (5), pressure with Eq. (6) and temperature profile with Eq. (7) for the gas inside the bubble without any further assumptions.
T 0b ðrÞ
... # " € b Rb R_ b R 1 5 ¼ q0 þ qr ð3c 2Þ 2 þ r4 0 21 Rb 40ðc 1Þkg Rb
þ CðtÞ:
2.2. Numerical integration of bubble wall motion
ð10Þ
The KM equation, Eq. (12), is usually integrated numerically, with being normalized by the appropriate physical variables. The
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radius is normalized by the equilibrium radius R0, while the velocity and pressure are related to constants and are given by u0 ¼ ðP1 =q1 Þ1=2 and P1, respectively. The constants for normalizing other physical quantities such as time, dynamic viscosity and surface tension were obtained from the condition that the KM equation is invariant after normalization [13,16]. The physical constants used in the non-dimensional equations are given as follows; Time: t 0 ¼ R0 =u0 . Thermal conductivity: k0 ¼ P1 u0 R0 =T 1 . Thermal diffusivity: a0 ¼ u0 R0 . Dynamic viscosity: l0 ¼ P 1 R0 =u0 . Surface tension: r0 ¼ P 1 R0 . In general, the characteristic frequency of the driving force f0 differs from the characteristic time of bubble wall motion 1/t0. Because the bubble wall motion described by Eq. (12) is also normalized by t0 with this normalization method, but the nondimensional time is recovered by 1/f0 in the numerical procedure, there is a time lag in bubble motion with respect to the characteristic time of the applied ultrasound f0, which is defined as [22]:
1 f0
s ¼ t0 :
ð14Þ
The bubble behavior under an applied ultrasound can be described correctly with the concept of the time lag [23]. For example, the expansion ratio calculated from the Rayleigh–Plesset (RP) equation with normalizing the applied frequency by 1/t0 is as high as 17.2 due to an artificial resonance [22], while the observed value is only 9.76 for an air bubble of R0 = 5.0 lm driven at fd = 12.926 kHz and PA = 1.33 atm [24]. This observed value, could be predicted with a time lag of 0.5 ls. The bubble radius-time curves obtained from the modified RP equation [25] and the KM equation with an appropriate relaxation time were found to be virtually the same except for the bouncing behavior [16], which suggests that the relaxational motion of the bubble against the applied ultrasound is very important in microbubble behavior.
q @ 2 Dq þ ðr uÞ ¼ 0 Dt r 2 @r
ð20Þ
where D/Dt is the material derivative. Using Eq. (17) the above continuity equation can be rewritten in terms of the number density of bubbles as:
DNb Nb @ 2 þ 2 ðr uÞ ¼ 0: Dt r @r
ð21Þ
Using Eqs. (18) and (19), one can obtain the velocity in the cluster from the continuity equation, Eq. (21). The velocity field in the cluster is given by:
u¼
1 1 dag 3 dRb r: þ 3 ag dt Rb dt
ð22Þ
With the help of Eq. (18), the velocity profile in the cluster simply reduces to
u ¼ ag
R_ b r: Rb
ð23Þ
The velocity profile given in Eq. (23) indicates that the cluster behaves in the same way that the bubbles in the cluster do. However, the degree of the coupled behavior is expected to be very weak because the value of ag is much smaller than unity, which is drastically different from the gas motion inside the spherical bubble oscillating under ultrasound, described by Eq. (5). To calculate the radiation pressure due to the pulsating bubble inside the cluster, one should solve the following momentum equation [6], which is valid when ag is much less than unity for bubble–liquid mixture [6]:
q
Du @p ¼ : Dt @r
ð24Þ
Using the velocity profile given in Eq. (23), one can explicitly obtain the quadratic radiation pressure field inside the cluster from the momentum equation, Eq. (24):
3. Velocity profile and radiation pressure field in a bubble cluster Consider a spherical bubble cluster containing n bubbles under ultrasound. Assume that the bubbles in the cluster are homogeneously distributed and pulsate together synchronously with the applied ultrasound, which was considered by Yasui et al. [9] and An [10]. The density of the bubble–liquid mixture of the cluster is approximately given by [26]:
q ¼ al ql þ ag qg ffi ql ð1 ag Þ
The continuity equation for the cluster with the average density given in Eq. (15) is as follows:
ð15Þ
where al and ag are the liquid and gas volume fraction, respectively, and ql is the liquid density. The gas volume, which depends on the instantaneous radius of the bubble, Rb, in the cluster, is given by
prad ðrÞ ¼ pcl þ
2 h
q ag rcl 2
Rb
€ b þ 2ð1 ag ÞR_ 2 Rb R b
i
"
2 # r 1 : r cl
ð25Þ
Using Eq. (16), the radiation pressure field can be approximated as:
prad ðrÞ ¼ pcl þ
q n Rb h 2
rcl
Rb Rb þ 2ð1 ag ÞR_ 2b
i
" 1
2 # r r cl
ð26Þ
ð16Þ
where pcl is the pressure at the cluster wall. In the above equation, the term 4pr 3cl N b =3 is replaced by the number of bubbles in the cluster, n. Taking b as the ratio of the radiation pressure at the cluster wall to the radiation pressure at the center, the radiation pressure field inside the cluster is:
where Nb is the number density of bubbles in the cluster. Assuming that the mass per unit volume in the cluster does not change with time [26], we have
" h 2 # i 1 Rb r 2 € _ : Rb Rb þ 2ð1 ag ÞRb 1 ð1 bÞ prad ðrÞ ¼ q n 2ð1 bÞ r cl r cl
ag ¼ 4pR3b Nb =3
ð17Þ
ð27Þ
where the subscript 0 denotes the initial state. Using Eqs. (15), (16) and (17), we have the relationship between ag and ag0 and the relationship between Nb and Nb0 from Eq. (17):
From Eq. (27), the average pressure field inside the cluster can be approximated as:
q =Nb ¼ q 0 =Nb0
ag ¼ ag0 ðRb =R0 Þ3 =f1 ag0 ½1 ðRb =R0 Þ3 g
ð18Þ
Nb ¼ Nb0 =f1 ag0 ½1 ðRb =R0 Þ3 g
ð19Þ
h i ð2 þ 3bÞ R q n b Rb R€ b þ 2ð1 ag ÞR_ 2b 10ð1 bÞ r cl h i € b þ 2ð1 ag ÞR_ 2 ¼ nq1 Rb R b
rad ¼ p
ð28Þ
where n ¼ ½ð2 þ 3bÞ=ð1 bÞ=10ð1 ag ÞnRb =r cl . The pressure field at the center of cluster, obtained from Eq. (27) with b = 2/3 is the same
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as the pressure field estimated by Yasui et al. [9] and An [10], which is given by:
3 R q n b Rb R€ b þ 2R_ 2b : 2 r cl
"
1=2 2 3 4p3 8kB T e e 1 1 hc pffiffiffi 2 k 3 pme 4pe0 me c3 h 3p B T e k 1 hc : exp k kB T e k
jk ¼ ni ne Z 2 V b
ð29Þ
The radiation pressure given in Eq. (28) obtained is valid only when continuum approach is possible for the bubbly mixture.
Ub dU b 3 Ub Rb U 2b 1 þ 1 2 Cb dt 3C b
1 U b Rb d Rb rad : PB Ps t þ p1 p ¼ 1þ þ q1 C b C b dt Cb
Using Eqs. (28) and (30) can be written more explicitly as:
Ub Ub dU b Rb þn 1þ Cb Cb dt
3 Ub 4 Ub U 2b þ nð1 ag Þ 1 þ þ 1 2 3C b 3 Cb
1 U b Rb d Rb PB Ps t þ p1 : ¼ 1þ þ q1 C b C b dt Cb
1
ð32Þ
where M eff ¼ 4pq
5. Radiation mechanism The hemispherical spectral radiance from a light source in any medium may be described as [28]:
expð2jk Rb Þ expð2jk Rb Þ 1 þ jk Rb 2j2k R2b
# ð33Þ
where jk is the absorption coefficient of photon and ebk is the hemispherical emissive power from a blackbody source, which is given by:
ebk ¼
2phc 5
2
k ðehc=kkB T 1Þ
:
ð34Þ
For the case of small absorption, Eq. (34) may be written within the limit of jkRb < 1 as: D
jk ¼
4 jk Rb 4pR2b ebk : 3
e2
4pe0
ne meff : pmcm2
ð38Þ
Here meff ¼ na mrtr , where meff is the effective frequency of electron-atom collisions, v is particle velocity, rtr is the transport scattering cross-section, and na is the number density of atoms. With this absorption coefficient, the spectral radiation due to electronatom collisions for one polarization may be written as:
ð39Þ The corresponding total emission for one polarization is given by:
" j ¼ ni ne V b
p2 8kB T e 6 pme
1=2
e2
4pe0
hrtr pme c
# 2 kB T e : hc
ð40Þ
The number density of electrons ne was obtained using the following the law of mass action for the simple-ionization reaction which is pressure dependent [18].
/2 1 /2
3 1 Rb .
"
ð37Þ
The mean absorption coefficient in a weakly ionized medium may be calculated using the following equation with an induced emission correction [29]:
ð31Þ
R_ b d 1 € b þ 3 R_ 2 ; Meff R_ 2b ¼ 2Meff Rb R dt 2 2 b Rb
D
" # 1=2 2 3 4p3 8kB T e e 1 1 pffiffiffi 3 pme 4pe0 me c3 h 2 3p2
" # 1=2 2
8kB T e e hrtr 1 hc 1: jk ¼ ni ne V b exp 3 kB T e k pme 4pe0 pme c k
The behavior of a bubble inside a bubble cluster may be estimated using Eq. (13) and the modified KM equation given in Eq. (31). The added terms due to radiation pressure in the KM equation may be considered as the change in the effective mass for the bubble [27] because the time rate change of the kinetic energy with the effective mass yields the key terms in the left hand side of Eq. (31):
jk ¼ 4pR2b ebk 1 þ
j ¼ ni ne Z 2 V b
jm ¼ na ne am;class ¼ ð30Þ
ð36Þ
The total emission of brake radiation can be obtained by integrating jk over all wave lengths. The result is found for one polarization and, is given by:
4. Multi-bubble sonoluminescence The motion of a bubble inside a cluster is affected by the radiation pressure field generated by the surrounding bubbles pulsating under an applied ultrasound. If the radiation pressure field is considered an imposed pressure, the KM equation may be written as,
#
¼
3=2 II 2pme hi 5=2 ðk T Þ exp B b 2 Pb Tb h
ð41Þ
where / is the degree of ionization and hi is the characteristic temperature for ionization. If electron excitation above the ground level is ignored, the value of II, which is related to the internal partition function due to the electron excitation becomes nearly constant. It is noted that departure from Saha equation for ionization should be recognized in such transient event [18] like sonoluminescence. The emission by electron-atom scattering, given in Eq. (40) becomes comparable to that by electron-ion scattering, given in Eq. (37) if the degree of ionization is less than 0.01, which corresponds to an electron temperature of approximately 20,000 K. For the light from the MBSL condition where gas temperature is approximately 10,000 K, the spectral radiation due to electron-atom collisions can be used. Once the time dependent temperature of the gas inside the bubble around the collapse point is known, one can calculate the total radiance using either Eq. (37) or Eq. (40) and obtain the full-width half maximum (FWHM) for the light emission. 6. Results of calculations and discussion
ð35Þ
The above equation represents the case of light emission with finite absorption or optical thickness of 4jkRb/3. In the case of a transparent emitter or jkRb 1, we rewrite Eq. (35) using the Rosseland mean free path [29] lR ¼ 1=jk and thereby obtain the spectral radiation for brake radiation due to ion-electron collisions for one polarization [30]:
The light from a cloud of argon bubbles under the MBSL condition with an applied ultrasound frequency of 20 kHz, is shown in Fig. 1. This picture was captured in a dark room by a 35 mm camera with a 30 min exposure time. The number of bubbles, which was estimated from the degradation rate of methylene blue in water under the MBSL condition, was approximately 3000 per cycle [31]. The size distribution of the bubble at the MBSL
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Fig. 1. Sonoluminescence from a bubble cloud in an applied ultrasound field with a frequency of 20 kHz and a power of 165 W (From Ref. [31]).
condition were measured by phase-Doppler technique in which He–Ne laser (633 nm) and two avalanche photodiodes as receiving optics where used. The measured volume which was focused on an expected MBSL region is approximately 103 cm3. About 3000– 6000 acquisition of individual bubbles yielded mean diameter. The average and Sauter mean diameters which were detected during the period of period of bubble expansion phase including the maximum size of argon bubbles under the MBSL condition were approximately 36.4, and 54.8 lm, respectively [4], so the average equilibrium radius in the MBSL condition was estimated to be approximately 5 lm. Based on these experimental data, a spherical bubble cluster containing 3000 micro bubbles with a radius of 3.5 mm was considered in this study. The equilibrium radius and ultrasound amplitude were 5 lm, and 1.4 atm, respectively, resulting in a maximum bubble radius of 25 lm in MBSL environment. Fig. 2 shows the bubble radius–time curves for a single bubble and a bubble in a cluster of 3000 bubbles in the same ultrasound
Fig. 2. Time-dependent bubble radius-time curve for a single bubble (dotted line) and a bubble (solid line) in a cluster of 3000 bubbles in water solution subjected to an ultrasound with a frequency of 20 kHz and an amplitude of 1.4 atm.
field with a frequency of 20 kHz, and an amplitude of 1.4 atm. The relaxation time of the bubble motion relative to the ultrasound filed was taken as 0.5 ls in this case. In this study, the radiation pressure at the cluster wall was taken as zero or b was taken as zero. The maximum bubble radius of a bubble in the cluster was considerably smaller than that of a single bubble in the same applied ultrasonic field. In fact, the maximum bubble radius of a bubble in the cluster decreased as the number of bubbles in the cluster increased, as shown in Table 1. Consequently, the minimum radius of a bubble in the cluster increased, and the gas pressure at the collapse point as well as the maximum temperature decreased as the number of bubbles in the cluster increased as clearly confirmed from Table 1. This may be due to an increase in the effective mass of the bubble [27] by the radiation pressure field retarding the compression process by the ultrasound field, causing the amount of time from equilibrium point to the collapse point from the equilibrium point to increase. No light emission can be expected when the number density of bubble in the cluster exceeds 28/mm3, which corresponds to the case of a 5030 bubbles in a cluster of 3.5 mm in radius. Above the limit number density of bubble, for example, 29/mm3, the average gas temperature reached around the collapse point is approximately below 4000 K so that the maximum spectral radiance due to the brake radiance cannot be measured. The temporal change of the gas temperature and the corresponding total emission with the electron-atom brake radiation for the bubble shown in Fig. 2 are shown in Fig. 3. Fig. 3 shows the maximum temperature calculated by Eq. (7), corresponding to light emission, occurs 400 ps prior to the collapse point where the bubble radius is at its minimum and the gas pressure is at its maximum as can be seen clearly in Fig. 4. On the other hand, the temperature calculated by Eq. (8) with the uniform pressure approximation inside the bubble has its maximum at the collapse point as can be seen from the slash-dot line in Fig. 3. Eq. (7), which includes the acceleration and deceleration of the bubble wall, produces a time of the light pulse occurrence that agrees with the experimental results, indicating that the light pulse bursts nanoseconds prior to the bubble collapse [32]. Note that the total emission was calculated by multiplying the number of bubbles in the cluster with the value estimated by Eq. (40). The calculated FWHM of the radiance is 210 ps, which is smaller than the observed value of 251.9 ps for a cluster of argon bubbles [4]. The pulse width for light emitted from a bubble in the cluster, however, is larger than that from a single bubble. The pulse width from a single bubble is approximately 150 ps [33]. As clearly seen in Fig. 3, the observed and calculated shape of the pulse width is Gaussian. The time duration above 7500 K is approximately 370 ps as confirmed in Fig. 3. The spectral radiance estimated from Eq. (39) by multiplying the number of bubbles inside the cluster at a temperature of 8000 K, which is the case of 3000 bubbles in a cluster, has broad peak near 600 nm as shown in Fig. 5. If the gas temperature inside the bubble increases, the broad peak shifts to a lower wavelength, the peak occurs at 430 nm at a gas temperature of 12,000 K for a bubble in a cluster of 1000 bubbles under the same ultrasound filed. On the other hand, the peak occurs at a higher wavelength of 750 nm when the gas temperature is 6800 K, which is the case of 4000 bubbles in a cluster, as can be seen in the insert in Fig. 5. Spectral characteristics with a broad peak around 450 nm for MBSL in a water solution were obtained by Didenko and Pugach [34], Ashokkumar and Grieser [35] and Wall et al. [36]. Recently, a similar radiation spectrum with a broad peak around 500 nm was observed for MBSL in Hg [37]. On the other hand, Didenko and Gordeychuk [38] observed a broad spectrum with a peak around 300 nm and with a prominent emission of the excited OH radical at 310 nm in a water solution saturated with noble gases.
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Table 1 Maximum bubble radius, bubble radius and gas pressure at the collapse point, maximum gas temperature and minimum bubble wall velocity for a bubble in a cluster containing different numbers of bubbles. The applied ultrasound frequency and amplitude are 20 kHz and 1.4 atm, respectively. Number of bubbles in cluster (n)
Maximum bubble radius (lm)
Bubble radius (lm) at collapse point
Gas pressure at collapse point (MPa)
Maximum gas temperature (K)
Minimum bubble wall velocity (m/s)
0 1000 2000 3000 4000 5000
37.60 29.23 26.05 23.24 21.71 21.45
0.529 0.561 0.575 0.583 0.607 0.627
5911 3637 2886 2518 2032 1645
21570 12590 10040 8830 7750 6910
1330 1020 886 800 693 569
Fig. 3. Temporal temperature change obtained from Eq. (7) (dotted line) and from Eq. (8) (slash-dot line) and total emission due to electron-atom collisions (solid line) for the case shown in Fig. 2. In the insert, the sonoluminescence pulse width measured from the argon bubble cloud is shown (Taken from Ref. [4]).
Fig. 4. Temporal change of bubble radius (solid line) and gas pressure (dotted line) obtained from Eq. (6) for the case shown in Fig. 2.
Experimental results that show different locations of maximum peaks in spectral radiation distributions may be due to clusters containing different numbers of bubbles.
Fig. 5. Spectral radiance from argon bubble clouds containing 1000 bubbles (solid line: left) and of 3000 bubbles (dotted line: right) evaluated at 12,000 and 8000 K, respectively. The insert shows the spectral radiance from argon bubble cloud containing 4000 bubbles evaluated at 6800 K.
7. Conclusions MBSL was studied hydro-dynamically to obtain the velocity profile and radiation pressure field inside a bubble cluster containing several thousand micro bubbles by solving the continuity and momentum equations for the bubbly mixture. The obtained velocity profile demonstrates that the cluster behaves similarly to the bubbles in the cluster fashion as the bubbles in the cluster even though the degree of the coupled behavior is expected to be very weak because the value of ag is much less than unity. The radiation pressure field derived here, however, is only valid when the continuum approach is possible for a bubbly mixture. We have shown that the maximum bubble radius attained by the bubble in the cluster is considerably smaller than the radius predicted for a single bubble under the same applied ultrasonic field with a frequency of 20 kHz and an amplitude of 1.4 atm. Additionally, the maximum bubble radius attained by the bubble in the cluster decreases as the number of bubbles in the cluster increases. As a result, the minimum radius of the bubble in the cluster increases, possibly due to the effective mass of the bubble increasing from the radiation pressure field effect. The maximum temperature occurs at 400 ps prior to the collapse point and thus produces light pulse occurrence time that agrees with the experimental results. The light pulse width and spectral radiance from a bubble in a cluster subjected to ultrasound were calculated by adding a radiation pressure term to the KM equation, and the values were compared to experimental values of the MBSL condition. There was close agreement between the calculated and observed values.
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Acknowledgements This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (20120843). References [1] N.A. Tsochatzidis, P. Guiraud, A.M. Wilhelm, H. Delmas, Determination of velocity, size and concentration of ultrasonic cavitation bubbles by phaseDoppler technique, Chem. Eng. Sci. 56 (2001) 1831–1840. [2] G. Servan, J.L. Laborde, A. Hita, J.P. Caltagirone, A. Gerard, On the interaction between ultrasound waves and bubble in mono- and dual- frequency sonoreactors, Ultrason. Sonochem. 10 (2003) 347–355. [3] P.B. Birkin, D.G. Offin, C.J.B. Vian, T.G. Leigton, Multiple observation of cavitation cluster dynamics close to an ultrasonic tip, J. Acoust. Soc. Am. 130 (2011) 3379–3388. [4] I. Ko, H. Kwak, Measurement of pulse width from a bubble cloud under multibubble sonoluminescence condition, J. Phys. Soc. Jpn. 79 (2010) 124401. [5] R. Omta, Oscillations of a cloud of bubbles of small and not so small amplitude, J. Acoust. Soc. Am. 82 (1987) 1018–1033. [6] L. D’agostino, C.E. Brennen, Linearized dynamics of spherical bubble clouds, J. Fluid Mech. 199 (1989) 155–176. [7] H.N. Oguz, A. Prosperetti, A generalization of the impulse and virial theorems with an application to bubble oscillation, J. Fluid Mech. 218 (1990) 143–162. [8] R. Mettin, I. Akhatov, U. Parlitz, C.D. Ohl, W. Lauterborn, Bjerknes forces between small cavitation bubbles in a strong acoustic field, Phys. Rev. E 56 (1997) 2924–2931. [9] K. Yasui, Y. Iida, T. Tuziuti, T. Kozuka, A. Towata, Strongly interacting bubbles under ultrasonic horn, Phys. Rev. E 77 (2008) 016609. [10] Y. An, Formulation of multibubble cavitation, Phys. Rev. E 83 (2011) 066313. [11] F. Dzaharudin, S.A. Suslov, R. Manasseh, A. Ooi, Effects of coupling, bubble size, and spatial arrangement of microbubble cluster in ultrasonic fields, J. Acoust. Soc. Am. 134 (2013) 3425–3434. [12] H. Kwak, J. Na, Hydrodynamic solutions for a sonoluminescing gas bubble, Phys. Rev. Lett. 77 (1996) 4454–4457. [13] H. Kwak, J. Na, Physical processes for single bubble sonoluminescence, J. Phys. Soc. Jpn. 66 (1997) 3074–3083. [14] J. Jun, H. Kwak, Gravitational collapse of Newtonian stars, Int. J. Mod. Phys. D 9 (2000) 35–42. [15] H. Lin, B.D. Storey, A.J. Szeri, Inertially driven inhomogeneities in violently collapsing bubbles: the validity of the Rayleigh–Plesset equation, J. Fluid Mech. 452 (2002) 145–162. [16] H. Kwak, H. Yang, An aspect of sonoluminescence from hydrodynamic theory, J. Phys. Soc. Jpn. 64 (1995) 1980–1992. [17] V. Kamath, A. Prosperetti, Theoretical study of sonoluminescence, J. Acoust. Soc. Am. 94 (1993) 248–260.
[18] W.G. Vincenti, C.H. Kruger, Introduction to Physical Gas Dynamics, Robert E. Krieger Publishing Co., New York, 1965. [19] M.I. Boulos, P. Fauchais, E. Pfender, Thermal Plasma, vol. 1, Plenum Press, New York, 1994. [20] J.B. Keller, M. Miksis, Bubble oscillations of large amplitude, J. Acoust. Soc. Am. 68 (1980) 628–633. [21] H. Kwak, S. Oh, C. Park, Bubble dynamics on the evolving bubble formed from the droplet at the superheat limit, Int. J. Heat Mass Transfer 38 (1995) 1707– 1718. [22] H. Kwak, J. Lee, S.W. Karng, Bubble dynamics for single bubble sonoluminescence, J. Phys. Soc. Jpn. 70 (2001) 2909–2917. [23] S.W. Karng, H. Kwak, Relaxation behavior of microbubbles in ultrasonic field, Jpn. J. Appl. Phys. 45 (2006) 317–322. [24] J. Jeon, J. Lee, H. Kwak, Possibility of upscaling for single bubble sonoluminescence at a low driving frequency, J. Phys. Soc. Jpn. 72 (2003) 509–515. [25] H. Kwak, S.W. Karng, Y.P. Lee, Rayleigh–Taylor instability on a sonoluminescing gas bubble, J. Korean Phys. Soc. 46 (2005) 951–962. [26] R.T. Lahey Jr., R.P. Taleyarkhan, R.L. Nigmatulin, I.S. Akhatov, Sonoluminescence and the search for sonofusion, Adv. Heat Transfer 39 (2006) 1–168. [27] R.E. Apfel, Acoustic cavitation in methods of experimental physics, in: P.D. Edmonds (Ed.), Ultrasonics, vol. 19, Academic Press Inc., 1981, p. 373. [28] S. Hilgenfeldt, S. Grossman, D. Loshe, A simple explanation of light emission in sonoluminescence, Nature 398 (1999) 402–405. [29] Ya. B. Zeldovich, Yu. P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamics Phenomena, Academic Press, New York, 1966. [30] J. Jeon, J. Na, I. Yang, H. Kwak, Radiation mechanism from a single bubble sonoluminescence, J. Phys. Soc. Jpn. 69 (2000) 112–119. [31] K. Byun, H. Kwak, Degradation of methylene blue under multibubble sonoluminescence condition, J. Photochem. Photobiol. A 175 (2005) 45–50. [32] B.P. Barber, S.J. Putterman, Light scattering measurements of the repetitive supersonic implosion of a sonoluminescing bubble, Phys. Rev. Lett. 69 (1992) 3839–3842. [33] R. Hiller, S.J. Putterman, K.R. Weninger, Time resolved spectra of sonoluminescence, Phys. Rev. Lett. 80 (1998) 1090–1093. [34] Y.T. Didenko, S.P. Pugach, Spectra water sonoluminescence, J. Phys. Chem. 98 (1994) 9742–9749. [35] M. Ashokkumar, F. Greiser, Sonophotoluminescence from aqueous and nonaqueous solutions, Ultrason. Sonochem. 9 (1999) 1–5. [36] M. Wall, M. Ashokkumar, R. Tronson, F. Grieser, Multibubble sonoluminescence in aqueous salt solutions, Ultrason. Sonochem. 6 (1999) 7–14. [37] A. Troia, D.M. Ripa, Sonoluminescence in liquid metals, J. Phys. Chem. C 117 (2013) 5578–5583. [38] Y.T. Didenko, T.V. Gordeychuk, Multibubble sonoluminescence spectra of water which resemble single bubble sonoluminescence, Phys. Rev. Lett. 84 (2000) 5640–5643.
Contents lists available at ScienceDirect
Ultrasonics Sonochemistry journal homepage: www.elsevier.com/locate/ultson
Hydrodynamic approach to multibubble sonoluminescence Shahid Mahmood, Yungpil Yoo, Jaekyoon Oh, Ho-Young Kwak ⇑ Mechanical Engineering Department, Chung-Ang University, Seoul 156-756, Republic of Korea
a r t i c l e
i n f o
Article history: Received 7 December 2013 Received in revised form 25 January 2014 Accepted 25 January 2014 Available online 2 February 2014 Keywords: Multibubble sonoluminescence Bubble cluster Radiation pressure Pulse width
a b s t r a c t The velocity profile and radiation pressure field of a bubble cluster containing several thousand micro bubbles were obtained by solving the continuity and momentum equations for the bubbly mixture. In this study, the bubbles in the cluster are assumed to be generated and collapsed synchronously with an applied ultrasound. Numerical calculations describing the behavior of a micro bubble in a cluster included the effect of the radiation pressure field from the synchronizing motion of bubbles in the cluster. The radiation pressure generated from surrounding bubbles affects the bubble’s behavior by increasing the effective mass of the bubble so that the bubble expands slowly to a smaller maximum size. The light pulse width and spectral radiance from a bubble in a cluster subjected to ultrasound were calculated by adding a radiation pressure term to the Keller–Miksis equation, and the values were compared to experimental values of the multibubble sonoluminescence condition. There was close agreement between the calculated and observed values. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction It is well known that in multi-bubble sonoluminescence (MBSL), several thousands of micro-bubbles are generated and collapsed synchronously with an applied ultrasound [1–3]. Recent measurements on the pulse width of a cloud of bubbles subjected to ultrasound using a time-correlated single photon counting technique indicated that the bubbles in a cloud collapse simultaneously to emit a light that is synchronous with the applied ultrasound [4]. A spherical bubble cloud subjected to harmonic far-field pressure excitation was investigated by Omta [5] and D’Agosta and Brennen [6]. They revealed that the natural frequency of the bubble cluster is always less than the natural frequency of the individual bubbles. Oguz and Prosperetti [7] investigated the interaction of two 100 lm bubbles subjected to an ultrasound with a moderate amplitude. Mettin et al. [8] considered the mutual interaction between two micro bubbles (R0 < 10 lm) in a strong acoustic field (Pa > 1 bar, fd = 20 kHz). They found that the strength and direction of the secondary Bjerknes forces due to the radiation generated by other bubbles differed from the forces predicted by linear theory. Yasui et al. [9] performed numerical simulations on a system of two bubbles and considered the interactions between n numbers of bubbles. They found that the expansion of a bubble during the rarefaction phase of ultrasound was strongly reduced by the presence of other bubbles in the cluster. They also obtained the ⇑ Corresponding author. Tel.: +82 28205278. E-mail address: [email protected] (H.-Y. Kwak). http://dx.doi.org/10.1016/j.ultsonch.2014.01.022 1350-4177/Ó 2014 Elsevier B.V. All rights reserved.
pressure field of the center of a cloud of similarly sized, homogeneously distributed bubbles which pulsated together with an applied ultrasound. An [10] obtained the radiation sound pressure from the other bubbles acting on a particular bubble in a cluster. He investigated the collective motion of similarly sized microbubbles in a cluster and found that the radiation pressure term added in the Keller–Miksis (KM) equation considerably suppresses bubble motion. Recently, Dzaharudin et al. [11] performed numerical simulations of a cluster of encapsulated microbubbles by adding the interaction term in the KM equation. They found that the oscillation amplitude of microbubbles that are close together was reduced for a given applied ultrasound power. In this study, MBSL is studied hydro-dynamically to obtain the velocity profile and radiation pressure field by solving the continuity and momentum equations for a spherical cluster containing numerous microbubbles. The calculated pulse width and spectral radiance for a bubble with the radiation pressure added in the KM equation are compared with the measured pulse width and spectral radiance values of the MBSL.
2. Single bubble behavior in an ultrasonic field 2.1. A set of analytical solutions for the Navier–Stokes equations The hydrodynamics related to the single bubble sonoluminescence phenomenon involves in solving the Navier–Stokes equations for the gas inside a bubble and the liquid adjacent to the
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bubble wall. The mass, momentum and energy equations for the gas inside a spherical bubble are given as;
@ qg 1 @ þ 2 q ug r2 ¼ 0; r @r g @t @P @ 1 @ ðqg ug Þ þ 2 qg u2g r2 þ b ¼ 0; @t r @r @r
qg C v ;b
DT b Pb d 2 1 d 2 r ug 2 ¼ 2 ðr qr Þ; r dr Dt r dr
ð1Þ
C¼ ð2Þ
ð3Þ
where r is the distance from center, qg is the gas density, ug is the gas velocity which obeys ug (Rb, t), the bubble wall velocity, Pb is the gas pressure, Cv,b is a constant volume specific heat and qr is the radial component of heat flux inside a bubble. A set of analytical solutions for the above conservation equations [12,13] is given as;
qg ¼ q0 þ qr ug ¼
R_ b r; Rb
Pb ¼ Pb0
ð4Þ ð5Þ
€ R 1 1 q0 þ qr b r 2 ; 2 2 Rb
TðrÞ ¼ T b ðrÞ þ T 0b ðrÞ
ð6Þ ð7Þ
where q0 is the gas density at bubble center and qr is the radial dependent gas density, which are given as q0 R3b ¼ const: and
qr ¼ ar2 =R5b ;respectively. The constant a is related to the gas mass inside a bubble and was taken as 5q0/(4p). Pb0 is the gas pressure at bubble center. The linear velocity profile describing the spatial in-homogeneities inside the bubble is a crucial ansatz for the homologous motion of a spherical object, which is encountered in another energy focusing mechanism of gravitational collapse [14]. The quadratic pressure profile given in Eq. (6), was recently verified by comparisons with direct numerical simulations [15]. The temperature profile due to the uniform pressure distribution Tb(r) is well known and is valid for a non-sonoluminescing gas bubble [16]:
2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 2 B 4 A A r 5 1 þ T b0 2g ðT bl T 1 Þ ; T b ðrÞ ¼ 1 þ A B B Rb
ð8Þ
where A and B are the are the coefficients of the temperaturedependent gas conductivity which has the form kg = AT + B [17], where g = (Rb/d)/(kl/B) and kl is the thermal conductivity of the liquid, d is thermal boundary layer thickness. The temperature at the bubble wall, Tbl in Eq. (9), can be obtained from Eq. (8) with a boundary condition of Tb Tb (Rb,t) = Tbl. The temperature is given as
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # B A A 2 2 T bl ¼ ð1 þ gÞ þ ð1 þ gÞ þ 2 T bo þ T þ gT 1 : A B 2B bo
The coefficient C(t) may be determined from the boundary con0 0 dition at the wall, kg dT b =dr ¼ kl dT l =dr;where Tl is the quadratic temperature distribution in the thermal boundary layer with a thickness d0 . That is,
h i ... 1 € b Rb þ Rb R2 ð3c 2ÞR_ b R b 20ðc 1Þ " # d0 5 Rb 5 q0 þ qr¼Rb þ 0 q0 þ qr¼Rb þ T 1 : 14 21 kl 2kg
The temperature distribution inside a bubble may be estimated 0 from Eqs. (7), (8), and (10) with appropriated values of d0 and kg , 0 where kg is gas conductivity in a dense plasma state [19]. The boundary layer thickness d0 may be determined from the relation 4pkl R2b ðT bl T 1 Þ=d0 ¼ P, where P is the power loss due to the brake radiation (bremsstrahlung) [12]. The temperature profile given in Eq. (10) yields a thermal spike when the acceleration of the bubble wall exceeds 1012 m/s2 [12] while the temperature distribution given in Eq. (8) provides a background temperature. The mass and momentum equation for the liquid adjacent to the bubble wall provides the well-known KM equation describing the motion for the bubble wall [20], which is valid when the bubble wall velocity does not exceed the speed of sound in the liquid. That is,
U b dU b 3 2 Ub Rb 1 þ Ub 1 2 C B dt 3C B
1 U b Rb d Rb PB Ps t þ P1 1þ þ ¼ q1 C B C B dt CB
ð12Þ
where Rb is the bubble radius, Ub is the bubble wall velocity, CB is the speed of sound in the liquid at the bubble wall, and q1 and P1 are the medium density and pressure, respectively. The liquid pressure on the external side of the bubble wall PB is related to the pressure inside the bubble wall Pb by PB = Pb 2r/Rb 4lUb/ Rb where r and l are the surface tension and dynamic viscosity of the liquid, respectively. The pressure of the driving sound field Ps may be represented by a sinusoidal function such as Ps = PAsin(xt) where PA is the driving sound amplitude, x = 2pfd and fd is frequency. For incompressible limit or Ub/CB ? 0, the KM equation reduces to the Rayleigh–Plesset equation. The mass and energy equation for the liquid provides a timedependent first order equation for the thermal boundary layer thickness d assuming a quadratic in temperature profile, which is given by [21],
"
" 2 # 2 # d 3 d dd 6a d 1 d dRb 1þ þ ¼ 2 þ Rb 10 Rb dt Rb 2 Rb d dt " 2 # 1 d 1 d 1 þ d 1þ 2 Rb 10 Rb T bl T 1
"
ð11Þ
dT bl dt
ð9Þ
ð13Þ
The temperature distribution given in Eq. (8) is valid until the characteristic time for the vibrational motion of the molecules is much less than the relaxation time for the translational motion of molecules [18]. The temperature rise and subsequent quenching due to bubble wall acceleration is given by:
where a is the thermal diffusivity of liquid. The above equation determines the heat flow rate through the bubble wall. The values for instantaneous bubble radius, bubble wall velocity and acceleration and the thermal boundary layer thickness obtained from Eqs. (12) and (13) are used to calculate the density with Eq. (4), velocity with Eq. (5), pressure with Eq. (6) and temperature profile with Eq. (7) for the gas inside the bubble without any further assumptions.
T 0b ðrÞ
... # " € b Rb R_ b R 1 5 ¼ q0 þ qr ð3c 2Þ 2 þ r4 0 21 Rb 40ðc 1Þkg Rb
þ CðtÞ:
2.2. Numerical integration of bubble wall motion
ð10Þ
The KM equation, Eq. (12), is usually integrated numerically, with being normalized by the appropriate physical variables. The
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radius is normalized by the equilibrium radius R0, while the velocity and pressure are related to constants and are given by u0 ¼ ðP1 =q1 Þ1=2 and P1, respectively. The constants for normalizing other physical quantities such as time, dynamic viscosity and surface tension were obtained from the condition that the KM equation is invariant after normalization [13,16]. The physical constants used in the non-dimensional equations are given as follows; Time: t 0 ¼ R0 =u0 . Thermal conductivity: k0 ¼ P1 u0 R0 =T 1 . Thermal diffusivity: a0 ¼ u0 R0 . Dynamic viscosity: l0 ¼ P 1 R0 =u0 . Surface tension: r0 ¼ P 1 R0 . In general, the characteristic frequency of the driving force f0 differs from the characteristic time of bubble wall motion 1/t0. Because the bubble wall motion described by Eq. (12) is also normalized by t0 with this normalization method, but the nondimensional time is recovered by 1/f0 in the numerical procedure, there is a time lag in bubble motion with respect to the characteristic time of the applied ultrasound f0, which is defined as [22]:
1 f0
s ¼ t0 :
ð14Þ
The bubble behavior under an applied ultrasound can be described correctly with the concept of the time lag [23]. For example, the expansion ratio calculated from the Rayleigh–Plesset (RP) equation with normalizing the applied frequency by 1/t0 is as high as 17.2 due to an artificial resonance [22], while the observed value is only 9.76 for an air bubble of R0 = 5.0 lm driven at fd = 12.926 kHz and PA = 1.33 atm [24]. This observed value, could be predicted with a time lag of 0.5 ls. The bubble radius-time curves obtained from the modified RP equation [25] and the KM equation with an appropriate relaxation time were found to be virtually the same except for the bouncing behavior [16], which suggests that the relaxational motion of the bubble against the applied ultrasound is very important in microbubble behavior.
q @ 2 Dq þ ðr uÞ ¼ 0 Dt r 2 @r
ð20Þ
where D/Dt is the material derivative. Using Eq. (17) the above continuity equation can be rewritten in terms of the number density of bubbles as:
DNb Nb @ 2 þ 2 ðr uÞ ¼ 0: Dt r @r
ð21Þ
Using Eqs. (18) and (19), one can obtain the velocity in the cluster from the continuity equation, Eq. (21). The velocity field in the cluster is given by:
u¼
1 1 dag 3 dRb r: þ 3 ag dt Rb dt
ð22Þ
With the help of Eq. (18), the velocity profile in the cluster simply reduces to
u ¼ ag
R_ b r: Rb
ð23Þ
The velocity profile given in Eq. (23) indicates that the cluster behaves in the same way that the bubbles in the cluster do. However, the degree of the coupled behavior is expected to be very weak because the value of ag is much smaller than unity, which is drastically different from the gas motion inside the spherical bubble oscillating under ultrasound, described by Eq. (5). To calculate the radiation pressure due to the pulsating bubble inside the cluster, one should solve the following momentum equation [6], which is valid when ag is much less than unity for bubble–liquid mixture [6]:
q
Du @p ¼ : Dt @r
ð24Þ
Using the velocity profile given in Eq. (23), one can explicitly obtain the quadratic radiation pressure field inside the cluster from the momentum equation, Eq. (24):
3. Velocity profile and radiation pressure field in a bubble cluster Consider a spherical bubble cluster containing n bubbles under ultrasound. Assume that the bubbles in the cluster are homogeneously distributed and pulsate together synchronously with the applied ultrasound, which was considered by Yasui et al. [9] and An [10]. The density of the bubble–liquid mixture of the cluster is approximately given by [26]:
q ¼ al ql þ ag qg ffi ql ð1 ag Þ
The continuity equation for the cluster with the average density given in Eq. (15) is as follows:
ð15Þ
where al and ag are the liquid and gas volume fraction, respectively, and ql is the liquid density. The gas volume, which depends on the instantaneous radius of the bubble, Rb, in the cluster, is given by
prad ðrÞ ¼ pcl þ
2 h
q ag rcl 2
Rb
€ b þ 2ð1 ag ÞR_ 2 Rb R b
i
"
2 # r 1 : r cl
ð25Þ
Using Eq. (16), the radiation pressure field can be approximated as:
prad ðrÞ ¼ pcl þ
q n Rb h 2
rcl
Rb Rb þ 2ð1 ag ÞR_ 2b
i
" 1
2 # r r cl
ð26Þ
ð16Þ
where pcl is the pressure at the cluster wall. In the above equation, the term 4pr 3cl N b =3 is replaced by the number of bubbles in the cluster, n. Taking b as the ratio of the radiation pressure at the cluster wall to the radiation pressure at the center, the radiation pressure field inside the cluster is:
where Nb is the number density of bubbles in the cluster. Assuming that the mass per unit volume in the cluster does not change with time [26], we have
" h 2 # i 1 Rb r 2 € _ : Rb Rb þ 2ð1 ag ÞRb 1 ð1 bÞ prad ðrÞ ¼ q n 2ð1 bÞ r cl r cl
ag ¼ 4pR3b Nb =3
ð17Þ
ð27Þ
where the subscript 0 denotes the initial state. Using Eqs. (15), (16) and (17), we have the relationship between ag and ag0 and the relationship between Nb and Nb0 from Eq. (17):
From Eq. (27), the average pressure field inside the cluster can be approximated as:
q =Nb ¼ q 0 =Nb0
ag ¼ ag0 ðRb =R0 Þ3 =f1 ag0 ½1 ðRb =R0 Þ3 g
ð18Þ
Nb ¼ Nb0 =f1 ag0 ½1 ðRb =R0 Þ3 g
ð19Þ
h i ð2 þ 3bÞ R q n b Rb R€ b þ 2ð1 ag ÞR_ 2b 10ð1 bÞ r cl h i € b þ 2ð1 ag ÞR_ 2 ¼ nq1 Rb R b
rad ¼ p
ð28Þ
where n ¼ ½ð2 þ 3bÞ=ð1 bÞ=10ð1 ag ÞnRb =r cl . The pressure field at the center of cluster, obtained from Eq. (27) with b = 2/3 is the same
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as the pressure field estimated by Yasui et al. [9] and An [10], which is given by:
3 R q n b Rb R€ b þ 2R_ 2b : 2 r cl
"
1=2 2 3 4p3 8kB T e e 1 1 hc pffiffiffi 2 k 3 pme 4pe0 me c3 h 3p B T e k 1 hc : exp k kB T e k
jk ¼ ni ne Z 2 V b
ð29Þ
The radiation pressure given in Eq. (28) obtained is valid only when continuum approach is possible for the bubbly mixture.
Ub dU b 3 Ub Rb U 2b 1 þ 1 2 Cb dt 3C b
1 U b Rb d Rb rad : PB Ps t þ p1 p ¼ 1þ þ q1 C b C b dt Cb
Using Eqs. (28) and (30) can be written more explicitly as:
Ub Ub dU b Rb þn 1þ Cb Cb dt
3 Ub 4 Ub U 2b þ nð1 ag Þ 1 þ þ 1 2 3C b 3 Cb
1 U b Rb d Rb PB Ps t þ p1 : ¼ 1þ þ q1 C b C b dt Cb
1
ð32Þ
where M eff ¼ 4pq
5. Radiation mechanism The hemispherical spectral radiance from a light source in any medium may be described as [28]:
expð2jk Rb Þ expð2jk Rb Þ 1 þ jk Rb 2j2k R2b
# ð33Þ
where jk is the absorption coefficient of photon and ebk is the hemispherical emissive power from a blackbody source, which is given by:
ebk ¼
2phc 5
2
k ðehc=kkB T 1Þ
:
ð34Þ
For the case of small absorption, Eq. (34) may be written within the limit of jkRb < 1 as: D
jk ¼
4 jk Rb 4pR2b ebk : 3
e2
4pe0
ne meff : pmcm2
ð38Þ
Here meff ¼ na mrtr , where meff is the effective frequency of electron-atom collisions, v is particle velocity, rtr is the transport scattering cross-section, and na is the number density of atoms. With this absorption coefficient, the spectral radiation due to electronatom collisions for one polarization may be written as:
ð39Þ The corresponding total emission for one polarization is given by:
" j ¼ ni ne V b
p2 8kB T e 6 pme
1=2
e2
4pe0
hrtr pme c
# 2 kB T e : hc
ð40Þ
The number density of electrons ne was obtained using the following the law of mass action for the simple-ionization reaction which is pressure dependent [18].
/2 1 /2
3 1 Rb .
"
ð37Þ
The mean absorption coefficient in a weakly ionized medium may be calculated using the following equation with an induced emission correction [29]:
ð31Þ
R_ b d 1 € b þ 3 R_ 2 ; Meff R_ 2b ¼ 2Meff Rb R dt 2 2 b Rb
D
" # 1=2 2 3 4p3 8kB T e e 1 1 pffiffiffi 3 pme 4pe0 me c3 h 2 3p2
" # 1=2 2
8kB T e e hrtr 1 hc 1: jk ¼ ni ne V b exp 3 kB T e k pme 4pe0 pme c k
The behavior of a bubble inside a bubble cluster may be estimated using Eq. (13) and the modified KM equation given in Eq. (31). The added terms due to radiation pressure in the KM equation may be considered as the change in the effective mass for the bubble [27] because the time rate change of the kinetic energy with the effective mass yields the key terms in the left hand side of Eq. (31):
jk ¼ 4pR2b ebk 1 þ
j ¼ ni ne Z 2 V b
jm ¼ na ne am;class ¼ ð30Þ
ð36Þ
The total emission of brake radiation can be obtained by integrating jk over all wave lengths. The result is found for one polarization and, is given by:
4. Multi-bubble sonoluminescence The motion of a bubble inside a cluster is affected by the radiation pressure field generated by the surrounding bubbles pulsating under an applied ultrasound. If the radiation pressure field is considered an imposed pressure, the KM equation may be written as,
#
¼
3=2 II 2pme hi 5=2 ðk T Þ exp B b 2 Pb Tb h
ð41Þ
where / is the degree of ionization and hi is the characteristic temperature for ionization. If electron excitation above the ground level is ignored, the value of II, which is related to the internal partition function due to the electron excitation becomes nearly constant. It is noted that departure from Saha equation for ionization should be recognized in such transient event [18] like sonoluminescence. The emission by electron-atom scattering, given in Eq. (40) becomes comparable to that by electron-ion scattering, given in Eq. (37) if the degree of ionization is less than 0.01, which corresponds to an electron temperature of approximately 20,000 K. For the light from the MBSL condition where gas temperature is approximately 10,000 K, the spectral radiation due to electron-atom collisions can be used. Once the time dependent temperature of the gas inside the bubble around the collapse point is known, one can calculate the total radiance using either Eq. (37) or Eq. (40) and obtain the full-width half maximum (FWHM) for the light emission. 6. Results of calculations and discussion
ð35Þ
The above equation represents the case of light emission with finite absorption or optical thickness of 4jkRb/3. In the case of a transparent emitter or jkRb 1, we rewrite Eq. (35) using the Rosseland mean free path [29] lR ¼ 1=jk and thereby obtain the spectral radiation for brake radiation due to ion-electron collisions for one polarization [30]:
The light from a cloud of argon bubbles under the MBSL condition with an applied ultrasound frequency of 20 kHz, is shown in Fig. 1. This picture was captured in a dark room by a 35 mm camera with a 30 min exposure time. The number of bubbles, which was estimated from the degradation rate of methylene blue in water under the MBSL condition, was approximately 3000 per cycle [31]. The size distribution of the bubble at the MBSL
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Fig. 1. Sonoluminescence from a bubble cloud in an applied ultrasound field with a frequency of 20 kHz and a power of 165 W (From Ref. [31]).
condition were measured by phase-Doppler technique in which He–Ne laser (633 nm) and two avalanche photodiodes as receiving optics where used. The measured volume which was focused on an expected MBSL region is approximately 103 cm3. About 3000– 6000 acquisition of individual bubbles yielded mean diameter. The average and Sauter mean diameters which were detected during the period of period of bubble expansion phase including the maximum size of argon bubbles under the MBSL condition were approximately 36.4, and 54.8 lm, respectively [4], so the average equilibrium radius in the MBSL condition was estimated to be approximately 5 lm. Based on these experimental data, a spherical bubble cluster containing 3000 micro bubbles with a radius of 3.5 mm was considered in this study. The equilibrium radius and ultrasound amplitude were 5 lm, and 1.4 atm, respectively, resulting in a maximum bubble radius of 25 lm in MBSL environment. Fig. 2 shows the bubble radius–time curves for a single bubble and a bubble in a cluster of 3000 bubbles in the same ultrasound
Fig. 2. Time-dependent bubble radius-time curve for a single bubble (dotted line) and a bubble (solid line) in a cluster of 3000 bubbles in water solution subjected to an ultrasound with a frequency of 20 kHz and an amplitude of 1.4 atm.
field with a frequency of 20 kHz, and an amplitude of 1.4 atm. The relaxation time of the bubble motion relative to the ultrasound filed was taken as 0.5 ls in this case. In this study, the radiation pressure at the cluster wall was taken as zero or b was taken as zero. The maximum bubble radius of a bubble in the cluster was considerably smaller than that of a single bubble in the same applied ultrasonic field. In fact, the maximum bubble radius of a bubble in the cluster decreased as the number of bubbles in the cluster increased, as shown in Table 1. Consequently, the minimum radius of a bubble in the cluster increased, and the gas pressure at the collapse point as well as the maximum temperature decreased as the number of bubbles in the cluster increased as clearly confirmed from Table 1. This may be due to an increase in the effective mass of the bubble [27] by the radiation pressure field retarding the compression process by the ultrasound field, causing the amount of time from equilibrium point to the collapse point from the equilibrium point to increase. No light emission can be expected when the number density of bubble in the cluster exceeds 28/mm3, which corresponds to the case of a 5030 bubbles in a cluster of 3.5 mm in radius. Above the limit number density of bubble, for example, 29/mm3, the average gas temperature reached around the collapse point is approximately below 4000 K so that the maximum spectral radiance due to the brake radiance cannot be measured. The temporal change of the gas temperature and the corresponding total emission with the electron-atom brake radiation for the bubble shown in Fig. 2 are shown in Fig. 3. Fig. 3 shows the maximum temperature calculated by Eq. (7), corresponding to light emission, occurs 400 ps prior to the collapse point where the bubble radius is at its minimum and the gas pressure is at its maximum as can be seen clearly in Fig. 4. On the other hand, the temperature calculated by Eq. (8) with the uniform pressure approximation inside the bubble has its maximum at the collapse point as can be seen from the slash-dot line in Fig. 3. Eq. (7), which includes the acceleration and deceleration of the bubble wall, produces a time of the light pulse occurrence that agrees with the experimental results, indicating that the light pulse bursts nanoseconds prior to the bubble collapse [32]. Note that the total emission was calculated by multiplying the number of bubbles in the cluster with the value estimated by Eq. (40). The calculated FWHM of the radiance is 210 ps, which is smaller than the observed value of 251.9 ps for a cluster of argon bubbles [4]. The pulse width for light emitted from a bubble in the cluster, however, is larger than that from a single bubble. The pulse width from a single bubble is approximately 150 ps [33]. As clearly seen in Fig. 3, the observed and calculated shape of the pulse width is Gaussian. The time duration above 7500 K is approximately 370 ps as confirmed in Fig. 3. The spectral radiance estimated from Eq. (39) by multiplying the number of bubbles inside the cluster at a temperature of 8000 K, which is the case of 3000 bubbles in a cluster, has broad peak near 600 nm as shown in Fig. 5. If the gas temperature inside the bubble increases, the broad peak shifts to a lower wavelength, the peak occurs at 430 nm at a gas temperature of 12,000 K for a bubble in a cluster of 1000 bubbles under the same ultrasound filed. On the other hand, the peak occurs at a higher wavelength of 750 nm when the gas temperature is 6800 K, which is the case of 4000 bubbles in a cluster, as can be seen in the insert in Fig. 5. Spectral characteristics with a broad peak around 450 nm for MBSL in a water solution were obtained by Didenko and Pugach [34], Ashokkumar and Grieser [35] and Wall et al. [36]. Recently, a similar radiation spectrum with a broad peak around 500 nm was observed for MBSL in Hg [37]. On the other hand, Didenko and Gordeychuk [38] observed a broad spectrum with a peak around 300 nm and with a prominent emission of the excited OH radical at 310 nm in a water solution saturated with noble gases.
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Table 1 Maximum bubble radius, bubble radius and gas pressure at the collapse point, maximum gas temperature and minimum bubble wall velocity for a bubble in a cluster containing different numbers of bubbles. The applied ultrasound frequency and amplitude are 20 kHz and 1.4 atm, respectively. Number of bubbles in cluster (n)
Maximum bubble radius (lm)
Bubble radius (lm) at collapse point
Gas pressure at collapse point (MPa)
Maximum gas temperature (K)
Minimum bubble wall velocity (m/s)
0 1000 2000 3000 4000 5000
37.60 29.23 26.05 23.24 21.71 21.45
0.529 0.561 0.575 0.583 0.607 0.627
5911 3637 2886 2518 2032 1645
21570 12590 10040 8830 7750 6910
1330 1020 886 800 693 569
Fig. 3. Temporal temperature change obtained from Eq. (7) (dotted line) and from Eq. (8) (slash-dot line) and total emission due to electron-atom collisions (solid line) for the case shown in Fig. 2. In the insert, the sonoluminescence pulse width measured from the argon bubble cloud is shown (Taken from Ref. [4]).
Fig. 4. Temporal change of bubble radius (solid line) and gas pressure (dotted line) obtained from Eq. (6) for the case shown in Fig. 2.
Experimental results that show different locations of maximum peaks in spectral radiation distributions may be due to clusters containing different numbers of bubbles.
Fig. 5. Spectral radiance from argon bubble clouds containing 1000 bubbles (solid line: left) and of 3000 bubbles (dotted line: right) evaluated at 12,000 and 8000 K, respectively. The insert shows the spectral radiance from argon bubble cloud containing 4000 bubbles evaluated at 6800 K.
7. Conclusions MBSL was studied hydro-dynamically to obtain the velocity profile and radiation pressure field inside a bubble cluster containing several thousand micro bubbles by solving the continuity and momentum equations for the bubbly mixture. The obtained velocity profile demonstrates that the cluster behaves similarly to the bubbles in the cluster fashion as the bubbles in the cluster even though the degree of the coupled behavior is expected to be very weak because the value of ag is much less than unity. The radiation pressure field derived here, however, is only valid when the continuum approach is possible for a bubbly mixture. We have shown that the maximum bubble radius attained by the bubble in the cluster is considerably smaller than the radius predicted for a single bubble under the same applied ultrasonic field with a frequency of 20 kHz and an amplitude of 1.4 atm. Additionally, the maximum bubble radius attained by the bubble in the cluster decreases as the number of bubbles in the cluster increases. As a result, the minimum radius of the bubble in the cluster increases, possibly due to the effective mass of the bubble increasing from the radiation pressure field effect. The maximum temperature occurs at 400 ps prior to the collapse point and thus produces light pulse occurrence time that agrees with the experimental results. The light pulse width and spectral radiance from a bubble in a cluster subjected to ultrasound were calculated by adding a radiation pressure term to the KM equation, and the values were compared to experimental values of the MBSL condition. There was close agreement between the calculated and observed values.
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