Journal of the Physical Society of Japan Vol. 75, No. 11, November, 2006, 114705 #2006 The Physical Society of Japan
Characteristics of Sonoluminescing Bubbles in Aqueous Solutions of Sulfuric Acid Ki Young KIM, Ki-Taek BYUN and Ho-Young KWAK Mechanical Engineering Department, Chung-Ang University, Seoul 156-756, Korea (Received August 8, 2006; accepted September 7, 2006; published October 25, 2006)
A new paradigm of sonoluminescence phenomena which displays alternating pattern of on/off luminescence pulse was observed recently in aqueous sulfuric acid solutions. A set of solutions of the Navier–Stokes equation for the gas inside a spherical bubble with heat transfer through the bubble wall permits to predict correctly characteristics of the sonoluminescing phenomena in the solutions. Calculation results of the minimum velocity of bubble wall, the peak temperature and pressure are in excellent agreement with the observed ones. Further, the calculated bubble radius–time curve displays alternating pattern of bubble motion as observed in experiment. The origin of sonoluminescence from gas bubble in sulfuric acid turns out to be blackbody emission with finite absorption. KEYWORDS: on/off pattern of luminescence, periodic doubling, sonoluminescence, sulfuric acid solutions DOI: 10.1143/JPSJ.75.114705
1.
analytically and how the heat transfer through the bubble wall is important in SL mechanism.
Introduction
Sonoluminescence (SL) phenomena associated with the catastrophic collapse of a gas bubble oscillating under an ultrasonic field1) have been extensively studied during the last 15 years or so for their exotic energy focusing mechanism.2) However, the radiation mechanism of sonoluminescence still remains unclear and the gas temperature and pressure at the collapse point have not yet been properly estimated. In fact, whether the spectrum of single bubble sonoluminescence under ultrasound frequency of kHz range in water is blackbody origin or bremsstrahlung one is still in dispute.3,4) The spectrum of single bubble sonoluminescence in water matches blackbody radiation with temperature in the range between 8000 – 20000 K.4) On the other hand, the spectrum from nanosize bubble under 1 MHz sound frequency was found to match thermal bremsstrahlung of 106 K plasma.5) Recent the observation of O2 þ emission from a sonoluminescing bubble in aqueous sulfuric acid solution6) strongly indicates also that the emission is bremsstrahlung from a hot plasma core and the bubble temperatures from the measured Ar emission are estimated to be 9,000 – 12,000 K. The SL from gas bubble in water is characterized by ten to hundred picoseconds flash, the bubble wall acceleration exceeding 1012 m/s2 ,7,8) and submicron bubble radius at the collapse point. On the other hand, recent investigation on the SL in aqueous sulfuric acid solution revealed rather longer flash width of ns, mild bubble wall acceleration of 1010 m/s2 and micron bubble radius at the collapse point under similar conditions of ultrasonic field,9) which opens a new paradigm of sonoluminescence phenomena. Further, the sulfuric acid system provides 3000 times brighter emission than that of the argon bubble in water even though the magnitude of the collapsing velocity is about 120 m/s,9) which is considerably less than the case of the sonoluminescing bubble in water. In this article, how the characteristics of SL in sulfuric acid solution is different from the one in water is discussed by treating the Navier–Stokes equations for the gas inside the bubble and the liquid adjacent to the bubble wall
E-mail:
[email protected]
2.
Theory
2.1
A set of solutions of the Navier–Stokes equations for the gas inside bubble The hydrodynamics related to the sonoluminescence phenomena involves in solving the Navier–Stokes equation for the gas inside bubble and the liquid adjacent to the bubble wall. The mass, momentum and energy equations for the gas inside the bubble with spherical symmetry are given as @g 1 @ þ 2 ðg ug r 2 Þ ¼ 0; r @r @t @ 1 @ @Pb ðg ug Þ þ 2 ðg u2g r 2 Þ þ ¼ 0; @t r @r @r DTb Pb d 2 1 d g Cv,b ðr ug Þ 2 ðr 2 qr Þ; ¼ 2 r dr Dt r dr
ð1Þ ð2Þ ð3Þ
A set of analytical solutions for the above conservation equations7,10) is given as ¼ 0 þ r R_b r; ug ¼ Rb
ð4Þ ð5Þ
€ Rb 2 1 1 Pb ¼ Pb0 0 þ r r ; 2 2 Rb TðrÞ ¼ Tb ðrÞ þ Tb0 ðrÞ
ð6Þ ð7Þ
where g R3b ¼ const. and r ¼ ar 2 =R5b . The constant a is related to the gas mass inside a bubble by a=m ¼ 5ð1 NBC Þ=4 with NBC ¼ ðPb0 R3b =Tb0 Þ=ðP1 R30 =T1 Þ, where Tb and T1 are the gas and the ambient temperatures, respectively. R0 is the equilibrium bubble radius, and the subscript 0 denotes the properties at the bubble center. The linear velocity profile showing the spatial inhomogeneities 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,11) and the quadratic pressure profile given in eq. (6), was verified recently by comparisons with direct numerical simulations.12)
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Using the definition of enthalpy, the internal energy equation for the gas can be also written as g Cp,b
DTb DPb 1 @ ¼ 2 ðr 2 qr Þ: r @r Dt Dt
ð8Þ
Eliminating D=Dtð¼ @=@t þ ug @=@rÞTb from eqs. (3) and (8), one can obtained the following heat flow rate equation inside bubble7,10)
DPb Pb @ 2 1 @ 2 ðr ug Þ 2 ðr qr Þ: ¼ 2 r @r Dt r @r
The temperature profile due to the uniform pressure distribution Tb ðrÞ which can be obtained by solving eq. (9) with the density and velocity profiles and uniform pressure distribution is well known:10)
2 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 B A A r 5; 1 þ Tb0 2 ðTbl T1 Þ Tb ðrÞ ¼ 41 þ A B B Rb where A and B are the are the coefficients in the temperature dependent gas conductivity having a form such as kg ¼ AT þ B, ¼ ðRb =Þ=ðkl =BÞ and kl is the thermal conductivity of the liquid and Tbl is the temperature at the bubble-wall. The thermal boundary layer thickness can be obtained from the Navier–Stockes equations for the liquid adjacent to the bubble wall. The gas temperature Tb0 and pressure Pb0 at the bubble center can be obtained from the ideal gas law 0 R3b ¼ const., one of the solution of the continuity equation give in eq. (1). The temperature distribution Tb0 ðrÞ due to the nonuniformity of the pressure distribution which induces the abrupt increase and subsequent decrease in the bubble wall acceleration near the collapse point was neglected in this study because the term is appreciable when the bubble wall acceleration exceeds 1012 m/s2 .7) Note that no additional equation of state is needed to obtain gas pressure inside the bubble in our formulation. 2.2
Governing equations from the Navier–Stokes equations for the liquid adjacent to the bubble wall The mass and momentum equation for the liquid outside bubble wall provides the well-known equation of motion for the bubble wall,13) which is valid until the bubble wall velocity does not exceed the sound speed of the liquid. That is Ub dUb 3 2 Ub þ Ub 1 Rb 1 2 CB dt 3CB 1 Ub Rb d Rb PB Ps t þ 1þ þ P1 ¼ 1 CB CB dt CB ð11Þ where Rb is the bubble radius, Ub is the bubble wall velocity, CB is the sound speed in liquid at the bubble wall, and 1 and P1 is the medium density and pressure. The liquid pressure on the external side of the bubble wall PB is related to the pressure inside the bubble wall Pb by PB ¼ Pb 2=Rb 4Ub =Rb where and are the surface tension and dynamics viscosity of liquid, respectively. The pressure of the deriving sound field Ps may be represented by a sinusoidal function such as Ps ¼ PA sin !t where PA is the driving sound amplitude, ! ¼ 2 fd and fd is frequency. The temperature distribution in the liquid layer adjacent to the bubble wall, which is important to determine the heat transfer through the bubble wall is assumed to be quadratic,14) such as
ð9Þ
T T1 ¼ ð1 Þ2 Tbl T1
ð10Þ
ð12Þ
Where ¼ ðr Rb Þ=. Such second order curve satisfies the following boundary conditions. TðRb ; tÞ ¼ Tbl ; and
@T @r
TðRb þ ; tÞ ¼ T1
¼0
ð13Þ
r¼Rb þ
The mass and energy equation for the liquid layer adjacent to the bubble wall with the temperature distribution given in eq. (12) provides a time dependent first order equation for the thermal boundary layer thickness.10) It is given by " 2 # 3 d 1þ þ Rb 10 Rb dt " 2 # 6
1 dRb 2 ð14Þ þ ¼ Rb 2 Rb dt " # 1 1 2 1 dTbl þ 1þ 2 Rb 10 Rb Tbl T1 dt where is the thermal diffusivity of liquid. The above equation determines the heat flow rate through the bubble wall. Instantaneous bubble radius, bubble wall velocity and acceleration and the thermal boundary thickness obtained from eqs. (11) and (14) provide density, velocity, pressure and temperature profiles for the gas inside the bubble without any further assumptions. No adjusted parameter is needed for calculation. In fact, the radius–time curve obtained by using the theory presented here for a nonsonoluminescing bubble of R0 ¼ 8:5 mm driven at PA ¼ 1:075 atm and fd ¼ 26:5 kHz exactly mimics the observed one.10) Assuming that the internal gas pressure is uniform and the gas behaves like ideal gas, Prosperetti et al.15) obtained the velocity profile inside bubble such as 1 @Tb 1 dPb ug ðr; tÞ ¼ ð 1Þkg r ð15Þ Pb 3 dt @r The above equation can also be obtained by integrating eq. (9) over the volume of the bubble for the case of uniform pressure. The time-dependent gas pressure inside the bubble can be obtained from the above equation by evaluating it at r ¼ Rb . That is
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" # dPb 3 @Tb ¼ ð 1Þkg Pb Ub Rb dt @r r¼Rb
ð16Þ
The internal energy given in eq. (8) can be rewritten with ideal gas law, Pb ¼ g RTb where R is gas constant, such as Pb @Tb @Tb dPb 1 @ @Tb r 2 kg þ ug ¼ 2 ð17Þ 1 Tb @t r @r @r dt @r Equations (15)–(17) replace the mass, momentum and energy conservation, respectively. The distributions for the gas velocity and temperature inside the bubble can be obtained by solving eqs. (15)–(17) simultaneously at a given
time with proper numerical method.16) In their study, the temperature distribution in the liquid adjacent to the bubble wall is assumed to be negligible so that the interface temperature is maintained to be equal to ambient one. Considerable computation time, however, is needed to obtain reasonable results with this formulation when the characteristic time of bubble evolution is ns range or below. The temperature distribution for the gas inside bubble with the boundary condition which Prosperetti et al.15) have chosen can also be obtained by solving eq. (9) with help of the set of solutions of the Navier–Stokes equation for the gas inside the bubble. That is
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 u 2 ( 2 2 ) 2 u A B A A r 5 Tb ðrÞ ¼ 41 þ t Tb0 þ 1 Tb0 þ 1 T1 þ 1 A B B B Rb As is well known, the above temperature distribution indicates the case of maximum heat transfer through the bubble wall regardless of the thermal properties of liquid. Usually a polytropic behavior of perfect gas17) was employed to obtain the gas pressure inside a uniformly compressed bubble. Pb ¼ Pb0
R3n 0 ðR3b a3 Þn
ð19Þ
The Rayleigh–Plesset equation with the polytropic relation, eq. (19) may determine the bubble behavior in liquid under ultrasound. For calculating the temperature the following relation with variable polytropic indexes of n which is related to the thermal diffusivity of gas and liquid and driving sound frequency may be employed.18) Tb R3ðn1Þ ¼ 3 0 3 n1 T1 ðRb a Þ
indexes in the study of sonoluminescence phenomena were discussed.17) 3.
Radiation Mechanism from Sonoluminescing Bubble
The hemispherical spectral radiance from the light source in any medium may be described as19) 2 j ¼ 4Rb eb ð21Þ expð2 Rb Þ expð2 Rb Þ 1 1þ þ 2 2 R b 2 Rb where is the absorption coefficient of photon and eb is the hemispherical emissive power from a blackbody source for the case of small absorption, eq. (21) may be written as in the limit Rb < 1,
j ¼
ð20Þ
For air bubble under ultrasound frequency of kHz range the polytropic index needed to calculated the temperature is about 1.3. However, any kind of polytropic approximation fails to account for the thermal damping effect due to finite heat transfer because Pb dV is a perfect differential.15) In fact, problems related using the Rayleigh–Plesset equation supplemented by the process equation with polytropic
ð18Þ
4 Rb 4R2b eb 3
ð22Þ
The above equation represents the case of light emission with finite absorption or optical thickness of 4 Rb =3. With wavelength dependent absorption coefficient , we can have the spectral radiance for bremsstrahlung due to electron–ion collisions or due to electron–atom collisions.20) For the case of the fully ionized gas, the following Rosseland mean free path lR may be used.
" 2 3 #1 pffiffiffi 2 1 4 3 8kB Te 1=2 3 e 2 3 kB Te lR ¼ ¼ : 4 3 me hme c 4"0 ni ne Z 2
ð23Þ
where me is the electron mass, Te is the electron temperature, h and kB are Plank and Boltzman constant, respectively and ni and ne are number density of ions and electrons in the bubble, respectively. Substituting eq. (23) into eq. (22), we have the spectral radiation for bremsstrahlung due to ion–electron collisions7) with a Gaunt factor at high frequency.21) It is " # pffiffiffi 3 3 43 8kB Te 1=2 e2 1 4 hc 2 pffiffiffi pffiffiffi GðT; Þ : ð24Þ j ¼ n i n e Z Vb 3 2 kB Te 8 3 me 4"0 me c h 3 3 where Vb is the bubble volume. The electron temperature Te is obtained by assuming a Mawellian electron velocity distribution. Equation (24) can be used for the volume emission of bremsstrahlung due to electron–ion collisions at higher gas temperature above 30,000 K. On the other hand, the mean absorption coefficient due to electron–atom scattering in a weakly ionized medium may be calculated using the following equation:22) 114705-3
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¼
e2 4"0
K. Y. KIM et al.
ne eff 2 1 ¼ 3 lA me c
ð25Þ
Here, eff ¼ na tr is the effective frequency of electron– atom collisions, na is the number density of atoms, is the electron velocity, and tr is the transport scattering crosssection. The emission by electron–atom scattering becomes comparable19) to that by electron–ion scattering if the degree of ionization is less than 0.01, which corresponds to an electron temperature of about 20,000 K.23) With , given in eq. (25), the spectral radiance in eq. (22) becomes " # e2 8kB Te 1=2 htr 1 n e n a Vb 2 4"0 me me c 3 ð26Þ j ¼ ; hc exp 1 kB T This equation may be used for the volume emission of bremsstrahlung due to electron–atom collisions. The total emission obtained from eq. (26) is more like blackbody emission from a volume radiation.20) 4.
Calcuation Results and Discussion
The calculated radius–time curve along with observed results for a xenon bubble with R0 ¼ 15 mm, driven by the ultrasonic field with a frequency 37.8 kHz and amplitude of 1.5 atm in aqueous solution of sulfuric acid is shown in Fig. 1. With air data for the thermal conductivity, the calculated radius–time curve which exactly mimics the alternating pattern of the observed result shows two different states of bubble motion. With xenon data, however, slight different pattern for the bubble motion was obtained. These calculation results imply that the bubble behavior, consequently the sonoluminescence phenomena depends crucially
80
60
50 Radius [µm]
Present theory (Air conductivity data) Present theory (Xenon conductivity data) Experimental results by Hopkins et al.
40 30 20
on the heat transfer in the gas medium as well as in the liquid layer and that the xenon bubble may contain a lot of air molecules. The Rayleigh–Plesset equation with polytropic relation, a conventional method17) used to predict the sonoluminescence phenomena cannot predict the two states of bubble motion as shown in the insert. The alternating pattern for the bubble motion may happen due to the entropy generation by the finite heat transfer through the bubble wall,24) which produces lost work: less entropy generation in one cycle having lower maximum bubble radius provides more expansion work to the bubble next cycle, while larger amplitude motion experiencing more entropy generation provides less expansion work to the subsequent motion. Further, the added mass due to the increase in medium density and heat transfer through the bubble wall reduce the expansion ratio, corresponding they reduce the peak temperature and pressure, considerably. The calculated minimum bubble radius for the light-emitting cycles, 4.6 mm is close to the observed value of 3.7 mm.9) As shown in Fig. 1 (insert), the calculated radius–time curve obtained by using the Rayleigh–Plesset equation with the polytropic relation does not show the alternating pattern because Pb dV is exact differential so that it does not produce entropy generation. Of course, the Rayleigh–Plesset equation with our temperature profile for the gas inside the bubble and the thickness of the thermal boundary layer, which are given eqs. (10) and (14), respectively provides the correct bubble radius–time curve. This implies that the Rayleigh–Plasset equation is valid perfectly for the bubble having lower maximum bubble wall velocity shown in Fig. 1. In Fig. 2, the bubble radius–time curves obtained with different wall temperatures which yield different temperature distribution given in eqs. (10) and (18), respectively for the bubble shown in Fig. 1 are given. No appreciable diffenrece in those bubble radius–time curves can be found. In fact, Hopkis et al.9) have shown that Prosperetti et al.’s formulation15) which employs the same governing equations but different bubble wall temperature from ours can reproduce the alternating pattern of the observed results.
10 0 10 20 30 40 50 60 Time [µs]
60 with Eq. (10) with Eq. (18)
40
50
Radius [µm]
Radius [µm]
0
20
40 30 20
0 0
10
20
30
40
50
60
10
Time [µ µs] 0 Fig. 1. (Color online) Theoretical radius–time curve along with observed one by Hopkins et al.9) for xenon bubble of R0 ¼ 15:0 mm at PA ¼ 1:50 atm and fd ¼ 37:8 kHz in sulfuric acid solution. The thermodynamic properties employed for 85% sulfuric acid solution are ¼ 1800 kg/m3 , Cs ¼ 1470 m/s, ¼ 0:025 Ns/m2 , ¼ 0:055 N/m, kl ¼ 0:40 W/mK, and Cp;l ¼ 1;817 J/(kg K).
30
40
50
60
Time [µs] Fig. 2. Calculated radius–time curves with different temperature distributions for the bubble shown in Fig. 1.
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Bubble wall velocity Bubble wall acceleration
14x10
120
60
9
100
100
10x109
50
8x10
9
0 6x109 4x109
-50
40 20 0
Radius [µm]
Velocity [m/s]
2
Acceleration [m/s ]
12x109
Present theory Experimental results
Radius [µm]
150
2x109
0
80
40 80 Time [µs]
60 40
-100 0 -150
-2x109 45.16 45.18 45.20 45.22 45.24 45.26 45.28 45.30
20
Time [µ µs] 0 0
Fig. 3. Calculated bubble wall velocity and acceleration near the collapse point for the bubble shown in Fig. 1.
40
60
80
100
Time [µ µs] Fig. 4. (Color online) Theoretical radius–time curve along with obserbed one by Hopkins et al.9) for xenon bubble of R0 ¼ 17:0 mm at PA ¼ 1:60 atm and fd ¼ 37:8 kHz.
70 60 50
Radius [µm]
Figure 3 shows the time-dependent bubble wall velocity and the variation of the bubble wall acceleration around the collapse point for the bubble shown in Fig. 1. The calculated magnitude of the minimum velocity at the collapse point for the light emitting cycles is about 115 m/s which is close to the observed velocity of 80 m/s. Whereas the maximum bubble wall velocity for non-light-emitting cycle is about 88 m/s, which is also close to the observed results of 60 m/s.9) The observed data for the bubble wall velocity were obtained from the minimum values calculated using the light scattering data for the time dependent bubble radius given in Hopkins et al.9) The calculated maximum bubble wall acceleration is about 1010 m/s2 . This value is smaller than the case of the sonoluminescing gas bubble in water by two orders of magnitude so that the gas pressure inside the bubble is almost uniform and the temperature increase due to the bubble wall acceleration is as small as 300 K. However, the magnitude of the minimum velocity calculated by Rayleigh–Plesset equation with the polytropic relation, which is about 900 m/s is much higher than the observed value. Similar alternating state of bubble motion as shown in Fig. 1 was observed by slight increase in the driving pressure amplitude and in the equilibrium radius,9) which can be also predicted correctly by the theory presented in this study. Figure 4 shows the bubble radius–time curve for a bubble with R0 ¼ 17 mm at PA ¼ 1:6 atm and fd ¼ 37:8 kHz. Quite different bubble behavior was obtained by the Rayleigh–Plesset equation with the polytropic relation for this case also as shown in the insert. However, the state of bubble motion shown in Fig. 4 yields light emission every cycle with weak intensity after strong intensity of luminescence. The calculated peak temperatures are 13,000 and 15,000 K for weak and strong emission, respectively. These states of bubble motion as shown in Figs. 1 and 4 are a bifurcated state obtained from the well known perioddoubling route25) from a state of bubble motion as shown in Fig. 5, which can be clearly confirmed from the frequency spectrum diagrams as shown in Fig. 6. The bubble shown in Fig. 1 has the same frequency spectrum diagram as the one
20
40 30 20 10 0 0
20
40
60
80
100
Time [µ µs] Fig. 5. Theoretical radius–time curves for xenon bubble of R0 ¼ 16:0 mm at PA ¼ 1:60 atm and fd ¼ 37:8 kHz.
for the bubble shown in Fig. 4. Figure 7 shows the calculated time-dependent bubble center temperature and the temporal emissive power with the average temperature for light-emitting cycle of the bubble shown in Fig. 1. The peak temperature calculated at the bubble center is about 8200 K, which is in excellent agreement with the observed value of 6000 – 7000 K. In fact, the average temperature at the collapse point is about 6000 K because considerable temperature drop occurs at the bubble wall as shown in Fig. 7 (inset). With the boundary condition of the wall temperature as the ambient liquid one, considerable temperature gradient inside the bubble can be found as shown in the insert. With this boundary condition which cannot account for the heat transfer in the liquid layer, the average temperature inside bubble becomes lower
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K. Y. KIM et al. Table I. Comparison between theoretical results and observed ones for the important parameters related to SL in sulfuric acid solutions. The observed minimum velocities were taken from the minimum values calculated using the light scattering data for the bubble radius–time curves.
Power Spectrum
1.0 0.8 0.6
Physical parameters
Observed results by Hopkins et al. (2005)
Our theoretical results
Minimum velocity at the collapse point for the nonlight-emitting cycle (m/s)
60
88
Minimum velocity at the collapse point for the light-emitting cycle (m/s)
80
115
1.0
Peak temperature at the collapse (K)
6000 – 7000
6000 (average) 8200 (at the center)
0.8
Peak pressure at the collapse (atm)
>1600
2800
0.6
FWHM of the luminescence pulse (ns)
10
21
0.4 0.2 0.0 0
1
2
3
4
5
Frequency [f/fd]
Power Spectrum
(a)
0.4 0.2 0.0 0
1
2
3
4
5
Frequency [f/fd] (b) Fig. 6. Frequency spectrum diagrams (a) for the bubble shown in Fig. 5 and (b) for the bubble shown in Fig. 4.
10000
3.0e-5 8000 6000 1000 4000 2000 0 950 0.0 0.2 0.4 0.6 0.8 1.0
6000
2.5e-5
2.0e-5
r/Rb
1.5e-5 Center temperature Total emission with average temperature
4000
2000
0 45.0
1.0e-5
5.0e-6
45.1
45.2
45.3
45.4
45.5
Total blackbody emission [W/sr]
Temperature [K]
8000
1050
Pressure [atm]
Temperature [K]
10000
0.0 45.6
Time [µ µs] Fig. 7. (Color online) Time dependent gas temperature at the bubble center and the corresponding total blackbody emission with the average temperature for the gas inside the bubble shown in Fig. 1. The temperature and pressure distributions at the collapse point are shown in inset where the dotted curve indicates the case with the temperature distribution given in eq. (18).
slightly. However, it has been reported that the liquid phase reaction zone adjacent to the bubble wall where extraordinary chemical reaction might take place have effective temperature of 1900 K.26) For the non-light-emitting cycle, the peak temperature calculated reduces to 6200 K while the calculated average gas temperature inside bubble becomes
about 4500 K. No explicit reason why light does not emit at this cycle was made. The pressure in the bubble is almost uniform as expected. Our calculated gas pressure at the collapse point for the argon bubble with R0 ¼ 13 mm driven at PA ¼ 1:4 bar and fd ¼ 28:5 kHz in sulfuric acid solution is about 2800 atm, which is also close to the lower bound value of observed result, 1600 atm.27) On the other hand, the Rayleigh–Plesset equation with the polytropic relation provides the peak temperature of 2500 K and the peak pressure of 9500 atm for the bubble shown in Fig. 1. Considerable overestimation in the gas pressure and underestimation in gas temperature are provided by the polytropic relation. In Table I, comparison between theoretical results and observed one is shown for the important physical parameters related to the SL for the case shown in Fig. 1. Equation (25) may also be employed when the optical path of the light is much greater than the wavelength of light so that surface emission prevails, which was realized in the laser-induced cavitation.28,29) For red light, the absorption coefficient is calculated to be approximately 60/cm with typical values of eff ¼ 1012 /s and ne ¼ 1019 /cm3 so that the optical thickness at the bubble collapse is approximately 0.018. With this constant value of the absorption coefficient the calculated full width at half maximum (FWHM) of the luminescence pulse is about 21 ns at the collapse point, which turns out to be in fair agreement with the observed value of 10 ns.9) Possible emission mechanisms for sonoluminescence observed so far are tabulated in Table II. All emission mechanism is possible for the sonoluminescence occurred in sulfuric acid solutions depending on the concentration amount of dissolved gas.9) The spectral radiances calculated with eq. (22) at the average temperature of 6000 K at the collapse point along with the observed value are shown in Fig. 8. Our calculated spectral radiance are in qualitative agreement with observed one. Further, our theory7,30) predicts correctly the flash width of micron bubble driven at 40 kHz and 1.45 atm and the peak temperature of 106 K achieved in a submicron bubble driven at 1 MHz and 4 atm.20) Close agreement between the calculated and observed values in every possible case
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Table II. Possible emission type from a single bubble sonoluminescence in liquids.
Emission type
Bubble radius (Rb ) compared to Medium SL the wavelength temperature flash of visible light (K) width ( )
Fluid medium
Rb
Tb > 20;000
ps
Water, water– glycerin mixture, alcohols, aqueous sulfuric solutions
Bremsstrahlung due to electron– atom collision
Rb
Tb < 15;000
ps
Water, water– glycerin mixture, alcohols, aqueous sulfuric solutions
Blackbody emission with finite absorption
Rb >
Spectral Radiance [W/nm/sr]
Bremsstrahlung due to electron– ion collisions
Tb < 10;000
ns
Aqueous sulfuric acid solutions
Calculated results Observed results 10-7
10-8
10-9 200
300
400
500
600
700
800
Wavelength [nm] Fig. 8. Spectral radiance due to blackbody emission with a constant value of absorption coefficient with the average temperature of 6000 K at the collapse point for the bubble shown in Fig. 1.
studied supports the theory presented in this paper as a reasonable model of sonoluminescence phenomena. 5.
Conclusion
The nonlinear behavior of an ultrasonically driven bubble and the sonoluminescence characteristics from the bubble in sulfuric acid solutions have been found to be correctly predicted by a set of solution of the Navier–Stockes equations for the gas inside the bubble with considering heat transfer through the bubble wall. The behavior of the bubbles turns out to be dependent crucially on the medium density and heat transfer through the bubble wall. The calculated bubble radius–time curves which mimic the observed ones show alternating pattern of bubble motion and yield the sonoluminescence characteristics from the gas bubbles in sulfuric acid solutions where diverse sonolumi-
nescence phenomena are possible to be occurred. These results cannot be obtained from the polytropic relation which have been used to predict the sonoluminescence phenomena. Acknowlegement This work has been supported by a grant from Electric Power Research Institute (EPRI) in U.S.A., under contract EP-P19394/C9578 and also supported by the Seoul R&BD program (2005). One (Ki Young Kim) of aurthors has been also supported by the second stage of BK21 program.
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