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A Theoretical Study of Sonoluminescence V A theoretical study of sonoluminescence V. Kamath and A. Prosperetti Departmentof MechanicalEngineering, The JohnsHopkins University,Baltimore, Maryland 21218 F. N. Egolfopoulos Departmentof MechanicalEngineering, University of SouthernCalifornia, Los Angeles,California 90089 (Received22 August 1991; revised16 October 1992; accepted11 March 1993) The productionof OH radicalsby dissociationof water vapor in oscillatingargon bubblesis studied theoretically to examine a possiblemechanism for the emissionof the 310-nm line observedin sonoluminescenceexperiments. Accurate modelsare usedfor the calculationof the temperaturefield in the gas and for the descriptionof the associatedchemical kinetics. Heat transferbetween the bubbleand the liquid is found to play a dominant role in the process.At the low exc.itation amplitudesconsidered, the bubbleradius is alsoan important parameter. PACS numbers: 43.35.Ei, 43.35.Sx INTRODUCTION confirms this strong dependence(Verral and Sehgal, 1988). Sonoluminescence,the weak light emissionassociated Very recently, Crum and Gaitan (Gaitan and Crum, with acousticcavitation activity, is an intriguing phenom- 1990; Gaitan, 1990) gave a remarkableexperimental dem- enonstill poorly understood.Several hypotheses have been onstration (implicit perhapsin the 1970 work of Saksena put forward to explain its origin (for a good review see and Nyborg; see also Crum and Reynolds, 1985) that so- Verral and Sehgal,1988), but essentiallyonly one--rooted noluminescencecan be producedby a singlestably oscil- in the large temperaturesand pressuresreached in the bub- lating air bubble.This discoveryopened the way to a much bles during collapse•is compatiblewith the availableex- more detailed experimentalstudy of the processthan had perimental evidence.Indeed, it is firmly establishedthat previouslybeen possible and a seriesof strikingnew results the luminescenceis emitted during the compressionphase is documentedin the recent papersof Putterman and co- of the sound field (Negishi, 1961; Gunther et al., 1959; workers (Barber and Putterman, 1991, 1992; Barber et al., Kuttruff, 1962; Gaitan and Crum, 1990; Roy and Gaitan, 1992; Lffstedt et al., 1993). 1991; Gaitan, 1990; Barber and Putterman, 1991; Barber In their experimental study, Taylor and Jarman et al., 1992), which rules out the triboluminescence (1970) showed that, when water is saturated with a noble (Chambers, 1936, 1937; Flosdorf et al., 1936), microdis- gas such as argon or xenon, the sonoluminescencespec- trum exhibits a strong peak at approximately 310 nm. charge (Frenkel, 1949; Sirotyuk, 1966), and mechano- Sehgalet al. (1980,1988) showedthis peak to be dueto the chemical (Weyl and Marboe, 1949; Weyl, 1951) theories. radiative transition of excited hydroxyl radicals to the Insensitivityto the liquid electricalconductivity (Negishi, ground state, 1961; Jarman, 1960) is incompatiblewith the ballo-electric mechanism(Harvey, 1939), and the anionic one (Degrois H20* ---,H + OH* --,H + OH + hv, (1) and Baldo, 1974) is similarly incompatiblewith the ob- where the excited H•.O moleculeis produced,e.g., by a served dependenceof the spectra upon the nature of the molecular collision. The excited H20 also has a lower dissolvedgas (Verral and Sehgal, 1988; Prudhommeand probability of a direct radiative decaywith an emissionin Guilmart, 1957; Gunther et aL, 1959; Margulis, 1969). the range between 270 and 290 nm. Other mechanisms Motivated by the early computationsof Noltingk and (reviewedin Verral and Sehgal,1988) havebeen proposed Neppiras (1950), Griffing and Sette (1955) were the first to account for the weaker broadband backgroundfound to proposethat the large temperaturesand pressuresat- experimentally.For air-saturatedwater the spectralook tained during the collapseof the oscillatingbubbles excited substantiallydifferent due to the combination of the OH chemicalreactions responsible for the emissionof light. In emissionswith thoseassociated with the nitrogenradicals 1963 Hickling presentedthe first numericalcalculations of (Verral and Sehgal,1988; Taylor and Jarman, 1970). a freely collapsingbubble in which the completethermo- From the point of view of the underlying chemistry, fluid dynamic behavior of the bubble's contents was ob- the simplestsonoluminescence process is that involvingno- tained from a direct numerical solution of the conservation ble gases, since in this case one only deals with the equationsfor mass,momentum, and energyin the gas.He hydrogen-oxygensystem, with the inert gas simply acting demonstratedthe large effectthat the thermal conductivity as a colliding agent to facilitate the unimoleculardecom- of the gashas on the peak temperaturesin the bubbleand positionreaction of the water vapor. For this reason,in the gave thus an explanationof the marked influenceof this presentpaper, we studythe formationof hydroxyl radicals physical property on the earlier experimentalresults of in stablyoscillating bubbles containing noble gases,prima- Proudhomme and Guilmart (1957). More recent work rily argon. For this purpose,we combinean earlier model 248 J. Acoust. Soc. Am. 94 (1), July 1993 0001-4966/93/94(1)/248/13/$6.00 ¸ 1993 AcousticalSociety of America 248 for the calculationof the gastemperature field (Prosperetti peratureincreases somewhat and so will thereforePv. The et al., 1988) with a detailedchemical kinetics description. gaspressure, however, increases by ordersof magnitudeso We find that hydroxyl radicalsare producedin a substan- that, again,pv has little impact on the dynamics.Hence, in tial numbereven at relativelylow excitationamplitudes. In the following,we shall disregardaltogether this contribu- this rangeof stablecavitation the processexhibits a strong tion in Eq. (4). dependenceon the bubbleradius, water temperature,and A major assumptionthat we now make is that the other parameters. internal pressureonly dependson time and is spatially While our study has been motivatedby the recent ob- uniform. This is justifiableat low excitationamplitudes servationsof stably oscillating,light-emitting air bubbles (Prosperettiet al., 1988;Prosperetti, 1991 ), but there is no previouslymentioned, the differencein the gascomposition assuranceas to its applicabilityto the presentconditions preventsus from attemptinga direct comparisonwith that and, a fortJori, to the experimentsof Gaitan and Crum work. Our results however seem to shed an interesting (1990) and Barberet al. (1992) at higherpressure ampli- light on somefundamental physical aspects of the process. tudes. If shock waves form in the gas, our results will With air, the ratio of specificheats of the gas is smaller considerably underestimate the actual dissociation rates than with a noble gasand heatingrates comparable to the and light emissionintensity. With the further assumption onesfound here would requirelarger driving pressure am- that the gas behavesaccording to the perfect gas laws, it plitudes.On the other hand, as far as the basicH20 mo- can be shownthat p is determinedby the equation (Pros- lecular decompositionprocess is concerned,one would not peretti, 1991) expectmajor qualitativedifferences with our findings.As noted above,the simultaneouspresence of differentexcited specieswould howeverstrongly affect the spectrumof the /5=•(y--1)K•r r •-ypl• , (5) emitted light. with the gas temperature T to be obtained by solving (Prosperetti, 1991) I. MODEL With one exceptionto be mentionedbelow, the model y--1T • +• (Y-1) K •rr -- • rt5•rr --t5 used to describethe bubble motion is essentiallythat of Prosperettiet al. (1988), which is briefly reviewedhere. =V. (KVT). (6) The bubble is assumedto oscillatespherically with a In theseequations y is the ratio of the gas heat capacities radius R (t) determinedby the Keller equation and K is the gas thermal conductivity. In Prosperettiet aL (1988) and Kamath and Prosper- (•)1-- R•-F•3(1--•c /•)/•2_c --•1 (l•+Rd) 14---c -c• [p•--pn]. etti (1989), the gas temperatureequation was solvedsub- (2) ject to the boundarycondition of constantsurface temper- ature, T(r=R(t), t)= Too, where T o• is the undisturbed Here time derivativesare denotedby dots,c is the speedof liquid temperature.This procedurewas justified with the soundin the liquid, and PL is the liquid density.The in- followingargument. If TL and K L denotethe liquid tem- stantaneousambient pressurein the liquid, including the perature and thermal conductivity,and phasechange ef- soundfield, is denotedby Pa- In this work we shall take fectsare unimportant,conservation of energyat the bubble PA=P oo( 1--e costot), ( 3 ) interfacerequires continuity of the heat flux whereP oois the undisturbedstatic pressure(taken as 1 8T 8T L bar•0.1 MPa--in the calculations that follow), to is the K•rr=KL Or ' (7) angular frequencyof the soundfield, and 6 its dimension- The temperaturegradients can be estimatedby introducing less pressureamplitude. The liquid pressurejust outside thermal layer thicknesses• and •L the bubblesurface, P B, is related to the internal gaspres- surep by 8T T6-- T s c•TL Ts-- Too 2• /• (8) p+po=pB-F•-+41a• , (4) where Ta is the temperatureof the bubble"core" and T s wherepo is the vapor pressure,a the surfacetension coef- the surface temperature. By using the estimates 6 ficient,and/• the liquid viscosity.At 20 øCthe vapor pres- .-• x/DAt, where At isa characteristictime (e.g., the oscil- sure of water is about 0.02 bars and thereforevery small lation period) commonto the gasand the liquid and D the comparedwith the partial pressureof the gasnecessary to thermal diffusivity, one
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