Modelling single-bubble sonoluminescence with chemical reactions and Coulomb interactions

Transcription

1 Modeing singe-bubbe sonouminescence with chemica reactions and Couomb interactions Ping He, Li Yuan Institute of Computationa Mathematics and LSEC Academy of Mathematics and Systems Science Chinese Academy of Sciences, Beijing 00080, China session: cavitation fied and sonouminescence September 6, 200 Abstract A refined hydro-chemica mode was appied to numerica simuation of singe-bubbe sonouminescence SBSL) by taking into account the processes of water vapor evaporation and condensation, heat and mass diffusion, and chemica reactions. The numerica resuts show significant water vapor dissociations but rather ow degrees of ionization. A widey accepted weaky ionized gas mode is then used to compute the ight emission. Contrary to earier predictions without chemica reactions, the cacuated ight spectra and puses are generay too sma to fit to experimenta data within stabe SBSL range. To aeviate this contradiction, the eectrostatic interactions of the ionized gas partices are taken into account, which are shown to ower the ionization potentias of gas species in the bubbe significanty, hence resuting in dramatic increase of ion popuations. Key words: singe-bubbe sonouminescence; Couomb interactions; owering of ionization potentia. Introduction Acoustic waves in iquids can cause cavitation. Cavitating bubbes undergo repeated cyces of growth and coapse in response to the acoustic waves. Under certain conditions, a singe bubbe trapped at the pressure antinode of an acoustic wave can emit a brief fash of ight during the vioent coapse of the bubbe, a phenomenon known as singe-bubbe sonouminescence [, 2]. SBSL has fascinated scientists from various fieds. Intriguing and extreme conditions were observed [2, ]. Many modes were proposed to expain the mechanism of the ight emission. The mode of therma bremsstrahung and recombination radiation from an opticay thin bubbe was most successfu as it predicted the widths, shapes and spectra of the emitted ight fairy we under certain hydrodynamic frameworks [4,, 6, 7]. However, chemica reactions were ignored in these predictions, whose infuence was found to reduce the temperature in the bubbe significanty [8]. A modification to the reaction rates was found to suppress vapor dissociations and Presented at Word Congress on Utrasonics/Utrasonics Internationa 0, Aug.29-Sept., Beijing, China.

2 raise the bubbe temperature to such an extent that the above ight emission mode recovered its pausibiity [9]. Yet a very recent study using a spatiay uniform pressure mode with chemica reactions predicted that the popuar therma bremsstrahung and recombination radiation mechanisms gave insufficient ight intensities compared with experimenta observations [0]. ther researchers found that therma bremsstrahung from a very sma emitting interior coud fit to the experimenta spectra more robusty []. In this paper, we use a refined fu hydrodynamic mode that accounts for the chemica reactions of water vapor mixture. As an initia step, the eectrostatic Couomb) interactions inside the bubbe are considered in a posteriori way. Numerica simuations indicated that the Couomb interactions reduce ionization potentias consideraby, resuting in significant increase in ion percentage. This resut is quite different from remarked in [0]. The ight spectra computed with and without the Couomb interactions are compared with a caibrated experiment [2]. 2 Mode The bubbe is assumed to be sphericay symmetric and is composed of mixture of nobe gas, water vapor and reaction products. The equations to be soved are the Navier-Stokes NS) equations couped with the Rayeigh-Pesset RP) equation, the water temperature equation and the mass concentration equation of the dissoved nobe gas in the surrounding water. Detaied formuations are given in Ref.[, 4]. A brief ist of the equations is as foows. The compressibe NS equations are written in the spherica coordinates: Q t + F = H + 2 F ν r 2 + M ν + S, ) where S is the chemica source terms. The weaky compressibe iquid fow outside the spherica bubbe is accounted for by a form of the RP equation that incudes first order terms in the Mach number M = Ṙ/C b and aows for variabe speed of sound in the water []: M)R R + 2 M ) Ṙ 2 = + M) T t + u T c t + u c [ H b ρ P s t + R C )] + R C b H b. 2) The equations for the water temperature T and for the mass concentration of dissoved nobe gas c take simiar form: λ = ρ C P r 2 r 2 T ), ) = D r 2 r 2 c ). 4) Eqs.)-4), together with appropriate boundary conditions, constitute the hydrodynamic-chemica mode, whose numerica soutions are sought for by time marching. The term S is computed by chemica kinetics, where ony a subset consisting of 8 eementary reactions is used, corresponding to the first eight ones used by Yasui [6]. The first 9 reactions of [6] with additiona species H 2, H 2 2 ) were aso tried but the resuting temperature was found ony a itte ower than that from the 8 reaction scheme. Nonequiibrium ionizations are not taken into account. Notice that the modification to the chemica equiibrium constant for a van der Waas gas as suggested by Toege et a. [9] is aso used in this study, see detai in [4]. Since the ight energy is very sma compared to the kinetic energy, the ight emission is postprocessed. This is done by using the ight emission mode of Higenfedt et a. [4]. This mode 2

3 accounts for the absorptions due to the free-free interaction of eectron and ions, free-free interactions of eectrons and neutra atoms, and bound-free ionization of aready excited atoms [4, 7]. The bound-bound absorption is ignored in this study. The we-known Couomb interactions exist for charged systems. A sonouminescing bubbe is thought to contain trace amounts of pasma, probaby at iquid-ike densities. Under such conditions, the effective ionization potentias tend to be owered [6]. Therefore, it is meaningfu to see how arge the effect of the Couomb interactions is for such constituents. According to the Debye-Hücke theory [8], the eectrostatic free energy F Coub is F Coub = 2e ) /2 π /2 Nj z 2 k B TV j). ) The tota free energy of the system is obtained by adding F Coub to the idea gas free energy. From the tota free energy one can obtain a modified Saha equation N j+ N e = Q ) j+q e Ij+ exp exp I ) j+ I j+, 6) N j Q j k B T k B T where I j+ = 2j + )e π/k B T) /2 n e + i 2 n i ) /2 describes a decrease in the ionization i potentia due to the Couomb interactions. We remark that the modified Saha equation 6) is used in a posteriori way from known number densities of atoms and other chemica radicas as cacuated from the hydrodynamic-chemica soutions, whose feedback infuence on the equation of state, hence on the hydrodynamics, is not considered for the time being. Numerica resuts The mode has foowing controabe parameters: the driving pressure ampitude P a and frequency f, the water temperature T, and the gas concentration dissoved in the water c. The ambient bubbe radius R 0 depends on above parameters. For comparison, we use neary the same set of parameters as in the experiment [2]: R 0 = 4. µm He) or. µm Xe), f = 42 khz, T = 296. K, P dissove = 0 torr He) or torr Xe), but P a is adjustabe. Fig. shows one snapshot of the number density distributions and the degrees of ionization. In Fig.a), it can be seen that the number densities of chemica products are in considerabe amounts. In Fig. b) it is seen that the degrees of ionization for the case with the Couomb interactions are much arger than those without the Couomb interactions. A threshod is set to prevent the ionization potentia from reducing to negative, which is refected in the sudden eveing of the curves in the inner zone for cases with the Couomb interactions. The reason that the ionization potentia tends to zero is probaby due to ack of a feedback mechanism to the hydrodynamics, as remarked in the end of Section 2. This wi be modified in future work. Fig. 2 shows the time history of the effective ionization potentia of atomic species. Note that the ionization potentias are reduced significanty around t = 0. Fig. shows comparison of the spectra radiances. It is seen that cacuated spectrum intensities with the Couomb interactions are coser to the experimenta data [2] under the same driving pressure within stabe SBSL range than without the Couomb interactions. However, the ong-waveength ight intensities are sti arger than the experimenta data for He bubbe. Simiar discrepancy aso existed for earier modes [6].

4 4 Concusions The fu hydrodynamic simuation of singe-bubbe sonouminescence indicates that the ight intensities are too sma to fit to experimenta data within stabe SBSL range. This resut is in agreement with recent one [0]. Furthermore, we studied the Couomb interactions of the ionized partices. We show that the ionization potentias of gas species are reduced significanty during a brief stage of the bubbe coapse, which eads to dramatic increase of ion percentage. This new resut awaits confirmation from further studies. Acknowedgements This work is supported by Nationa Natura Science Foundation of China G072089, G047602) and State Key Program for Basic Research G ). References [] B. Barber, R. Hier, R. Löfstedt, S. Putterman, and K. Weninger, Phys. Rep. 28, 6 997). [2] M. Brenner, S. Higenfedt, and D. Lohse, Rev. Mod. Phys., 742), ). [] R. Taeyarkhan et a., Science 29, ). [4] S. Higenfedt, S. Grossmann, and D. Lohse, Phys. Fuids, 8 999). [] W. Moss et a., Phys. Rev. E 9, ). [6] D. Hammer and L. Frommhod, J. Mod. pt. 48, ). [7] P. Burnnet et a., J. Phys. B: At. Mo. pt. Phys. 4 6), L 200). [8] B. Storey and A. Szeri, Proc. Roy. Soc. London A 46, ). [9] Toege R., Higenfedt S., and Lohse D., Phys. Rev. Lett. 88, ). [0] C. Ying, Y An and C.Xie, J. Phys. D: Appy. Phys. 8, ). [] C. Camara, S. Putterman, and E. Kiriov, Phys. Rev. Letts. 92, ). [2] G. Vazquez, C. Camara, S. Putterman, and K. Weninger, pt. Lett. 26, 7 200). [] C. Ho, L. Yuan, M. Chu, P. Leung, and W. Wei, Phys. Rev. E 6, ). [4] L. Yuan, Sonochemica effects on singe-bubbe sonouminescence, phyics/0004. [] A. Prosperetti and A. Lezzi, J. Fuid Mech. 68, ). [6] Yasui K., Phys. Rev. E 6, ). [7] Y. Zedovich and Y. Raizer, Physics of shock waves and high-temperature hydrodynamic phenomena, Vo. I, ed. W. D. Hayes and R. F. Probstein Academic Press, New York, 966). [8] S. Eiezer, A. Ghatak and E. Teer, An Intorduction to Equations of State: Theory and Appications Cambridge University Press, Cambridge, 986). 4

5 7 a) 0 0 b) number density 0 moe/m ) H H H 2 2 H 2 degree of ionization H + w. Cou. + w. Cou. H + wt.cou. + wt. Cou r µm) r µm) Figure : The spatia profies of number densities for moecuar species a) and degrees of ionizations b) at the time of minimum bubbe radius t = 0 t min = µs) for He bubbe at P a =.4 atm, R 0 = 4. µm. The degree of ionization is computed using the Saha equation. effective ionization potentia ev) He H radiance W/nm) He Xe wt.coub.pa=.4 atm) He wt.coub.pa=.4 atm) Xe Xe wt. Coub. Pa=. atm) He wt.coub.pa=2.0 atm) time reative to minimum radius ps) Figure 2: Time history for the effective ionization potentias of atomic species in He bubbe. Each ionization potentia is averaged over the whoe bubbe. P a =.4 atm, R 0 = 4. µm. t = 0 ps corresponds to the time of minimum radius t min = µs) waveength nm) Figure : Spectra radiance of the SL ight from bubbes of Xe and He in water. The squares and trianges are experimenta spectra of Xe and He bubbes [2]. The soid ines are fittings with the Couomb interactions at P a =.4 atm Xe),.4 atm He), the dashed are without them for the same P a, and the dotted ines are without them for P a out of stabe SBSL range.