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 then obtains from (7) maintainthe bubblein equilibrium.As the bubbleexpands, this partial pressurefalls while po remainsessentially con- r•-r• (KCpGpG•1/2 stant (Plesset and Prosperetti, 1977). However, by the Tq--Ts -- •KLCpL•] ' (9) timep =po, the pressurein the bubbleis negligibleanyway whereCp is the specific heat per unit mass. The term in the and the motionis governedby pAand inertialeffects. When right-handside is typicallyof the orderof 10-3 or 10-2 , the bubble contracts, as will be seenbelow, its surfacetem- which showsthat the majority of the temperaturedrop
249 d. Acoust.Soc. Am., Vol. 94, No. 1, July 1993 Kamathet aL: A theoreticalstudy of sonoluminescence 249 TABLEI. Thechemical kinetic model used in thecalculations. The constants kf andko are used in thecalculation of theforward and backward reaction rates,respectively, according toEqs. (15) and (17). Activation energies Ea, Eo in calories per mole, concentrations inmoles per cm 3. The notation used in thecolumns labeled Af andA o is such that, e.g., 1.92D14= 1.92 X 1014,etc.
No. Reaction Af fa cal/mol A0 fit, cal/mol 1 H+O2=O+OH 1.92D14 0.0 16440 7.18Dll 0.36 --679 2 O+ H2=H +OH 5.08D4 2.67 6292 2.64D4 2.65 4462 3 OH+H2=H+H20 2.18D8 1.51 3430 1.02D9 1.51 18620 4 OH+OH=H20+O 2.1D8 1.4 397 2.21D9 1.4 16628 5 H2+M=H+H+M 4.58D19 --1.4 104400 2.45D20 --1.78 960 Coef. H2/2.5/H20/16.0/ 6 O-FO-3-M=O2+M 6.17D15 --0.5 0 1.58D17 --0.5 118182 Coef. H2/2.5/H20/16.0/ 7 O+H+M=OH+M 4.72D18 -- 1.0 0 4.66D17 --0.65 101660 Coef. H20/5.0/ 8 H+OH+M=H20+M 2.25 D22 --2.0 0 1.96D22 -- 1.62 118600 Coef. H2/2.5/H20/16.0/ 9 H+O2+M=HO2+M 2.00D15 0 -- 1000 2.46D15 0 48290 Coef. H2/2.5/H20/16.0/ 10 HO2 + H = H 2+ 02 6.63 D13 0 2126 2.19 D13 0.28 56420 11 HO2+H=OH+OH 1.69D14 0 874 1.08D11 0.61 36220 12 HO2+O=OH+O2 1.81 D13 0 --400 3.1D12 0.26 51832 13 HO2 + OH = H20 + 02 1.45D16 -- 1.0 0 2.18D16 --0.72 69181 14 HO 2+ HO 2= H202 -+-02 3.0D12 0 1387 4.53 D14 --0.39 39140 15 H202+M=OH+OH+M 1.2D17 0 45500 9.D11 0.90 --6062 Coef. H2/2.5/H20/16.0/ 16 H202 + H = H20 + OH 3.2D14 0 8960 1.14D9 1.36 75870 17 H202+H=H2+HO 2 4.82D13 0 7948 1.41 D11 0.66 24480 18 H202+O=OH+HO 2 9.55 D6 2 3970 4.62 D3 2.75 18435 19 H202 + OH = H20 + HO2 1.00D13 0 1800 2.8D13 0 32790
from T o to Toooccurs from the bubblecore to the surface neon we take A=0.0159 W/mK, B----10-4 W/mK 2 and that T s cannotdiffer greatly from Too.In spiteof this which fits the data over the interval from 90 to 1600 K. argument,in order to avoid any ambiguityderiving from the assumptionof a constantsurface temperature, in this paperwe shall alsocalculate the temperaturefield in the II. CHEMICAL KINETICS liquid. For this purposewe needto solvethe equation As stated in the Introduction, there are indications that the strong310-nm peak observedin sonoluminescence pLCpL['•'-q-r 2 •rr)--V' (K L VTL), (10) spectrais dueto the radiativedecay of the hydroxylradical from an excited state. The simplestsituation to study is in the domain R (t)
250 d. Acoust. Soc. Am., Vol. 94, No. 1, July 1993 Kamath et al.: A theoretical study of sonoluminescence 250 For eachspecies appearing in Table I, e.g.,H, we write III. NUMERICAL METHOD a rate equationof the form It is convenient for numerical work to deal with a fixed d[H] rather than a moving boundary.We achievethis objective _ • (production)--• (destruction), (13) dt by introducing the new radial variable y=r/R(t). (19) where [H] is the concentrationof atomichydrogen and the first sum containsthe contributionsof all reactionsproduc- It is also convenient to transform the unbounded domain ing H, and the secondone that of all the reactionsconsum- occupiedby the liquid into a bounded one, and for this ing H. For example,for the first reaction in the table, the purposewe use the algebraictransformation forward processcorresponds to a consumptionof H and thus contributes the term 1/•'= 1+ (y- 1)/L, (20) whereL is chosento be proportionalto the diffusionlength kœ[H][O2], (14) in the liquid, to thesecond summation in (13). Herekf is theforward L=I( DL/CORo2)1/2, (21) reaction rate for reaction No. 1 and is parametrizedin the modified Arrhenius form in whichDL=KL/CœLpL is the liquidthermal diffusivity and l a numerical constant for which the value •=20 has been chosen for the calculations after some trials. With the (15) kf =Af Tl•aexp( --Ea/•2 T), mapping (20) the bubbleboundary r = R (t) is transformed where • is the universalgas constantand the constants to •--1 while infinity correspondsto •=0. Thus, with the Af, 15a, and Ea are givenin the first line of the table. use of Chebyshevpolynomials, we gain an excellentreso- Reaction No. 1 in the backward direction, on the other lution in the thermal boundary layer adjacentto the bub- ble. hand, producesH, and thereforecontributes to the first summation in (13) the term The numerical algorithm that has been used to solve the partial differential equations(6) and (10) is basedon k,[O] [OH], (16) a collocationform of the pseudospectralmethod which is extensivelydocumented (Gottlieb and Orszag, 1977; Ca- with the backward reaction rate nuto et aL, 1988). An earlierapplication to a similarprob- lem in gasbubble dynamicsmay be found in Kamath and kb=AbTt•bexp( --Eb/• T). (17) Prosperetti (1989) which also contains a more detailed descriptionof the method. The temperaturefields are first For the three-bodybackward reaction No. 5 the contribu- approximated by their projectionsonto a basis of Cheby- tion to the last term of (13) is shevpolynomials. In view of the conditionsof zero gradi- ent at the bubblecenter (y=0) and at infinity (•=0), only Ckb[H]2[M], (18) the even polynomialsT2• are required.Thus, the temper- ature field in the liquid is approximatedby whereM standsfor any third molecule.The multiplicative coefficientC indicatedin the secondline of the table entry TL M is 1 if M is a noblegas atom, but shouldbe takenas 2.5 for O=•-•=oM=k=0• blc(k) T2/c(•). (22) collisionsinvolving an H 2 moleculeand as 16 for collisions with an H20 molecule.The sameconvention applies to all The liquid energyequation (10) may be compactlywritten reactionsinvolving M in the table.
The schemeof Table I has been partially validated 0t --•[•], (23) from hydrogen flame studies (Egolfopoulosand Law, 1990) as well as shock-tubeand reactor-typeexperiments where,in termsof •, the operator• is givenby (Yetter et al., 1991). The backward rates indicated have been determined from the forward rates and thermody- namic equilibrium by using the thermodynamicdata of Kee et al. (1987). It shouldbe emphasized,however, that the unimolecular-decompositionreactions, as describedin Table I, have beentested experimentally only up to about --•g2 r2DL[•+• Y--• O• (24) 30 bars. For higher pressuresa gradualmodification of the with y given by (20) as a function of •. rates is expectedaccording to the Lindemann mechanism By writing (23) at the Gauss-Lobatto collocation (Gardiner and Troe, 1984) which is basicallyan asymp- points totic approach toward the high-pressurelimit. This •i=cos(•j/2M), j =0,1,...,M, change,however, is not expectedto affectqualitatively the physicsand conclusionsdrawn below, since its net effect one obtainsa systemof ordinary differentialequations for will only be to decreasesomewhat the rates from the values the unknowncoefficients b•, k = 0,1,...,M, of the expansion used in the presentcalculation. (22)
251 d. Acoust.Soc. Am., Vol. 94, No. 1, July 1993 Kamathet aL' A theoreticalstudy of sonoluminescence 251 25 • dbk Z T2k(•'j)•-=•'•[0M(•j)], (25) 24 I I I I31 I 39' ' / k=0 23 for j = 1,2,...,M; the first point •0 = 1 is the bubblesurface 2.2 at which the boundaryconditions must be applied. _ A similar procedureis used for the gas temperature •2.0 equationafter the introductionof the auxiliaryvariable 1.9
1.8
K(T')dT', (26) 1.7 Too - Ro/Rr•• 16 - 0.1 O.Z 0.3 where!3= D(poo+ 2a/R 0) is a normalizationconstant ex- 15 10 •s •o •s 3o s5 40 4s so pressedin termsof the undisturbedgas thermal diffusivity I•o(•m) and pressure.Now the expansionanalogous to (22) is
N FIG. 1. Normalized maximum radiusof an argon bubbleduring steady •'•.rN= • a2kT2k(Y), (27) oscillationsat 21 kHz and 0.93 relativepressure amplitude. The numbers k=0 alongthe curveindicate the valuesof the radiuscorresponding to the from which a form similar to (25) is obtained upon sub- resonancepeaks. The curve has been traced by linear interpolation throughthe computedresults (squares). The radiusRre s of a bubble stitution into (6). resonatingat this frequencyis 150 btm. Since the previousapproach reduces the problem to the solutionof a systemof ordinary differentialequations, it is usefulto recastthe boundaryconditions into a similar cal kinetic model for air rather than argon would require form. By differentiatingwith respectto time Eq. (11 ) we many more reactionsto accountfor the nitrogenchemis- find try. Becauseof the simultaneouslight emissionfrom the nitrogen radicals,the spectrumwould also be different. 0 • da• •t dbk (28) Thermally, however,the primary differencebetween air K(Ts) k-0 dr=Tøø k=0• dt ' and a noblegas would be causedby the much larger adi- 5 Similarly, time differentiationof (7) gives abaticindex of a monomoleculargas, • ascompared to -•. Becauseof this, the argonbubble heats up moreeffectively than an air one would and, therefore, one would expect O • (2k)2 (29) •=o •-=KœToo•=o• (2k) 2dt ' radicalproduction to occurat a somewhatlower pressure amplitude. The mathematical structure of the problem consists In a typical sonoluminescenceexperiment, when the now of the M differentialequations (23), the analogousN soundfield is turned on, no bubblesare present.Bubbles differentialequations for the gas temperature,the bound- grow from a small size by rectifieddiffusion and coales- ary conditions(28), (29), the Keller equation(2), the cencewith other bubblesuntil they become"a&ive." The pressureequation (5), and the chemicalkinetics equations interestingregion is thereforethat belowthe resonantsize, (13) written for each speciesappearing in Table I. Their and indeedthis is the regionthat has beeninvestigated in numericalsolution is effectedby usingGear's method for the recent experimentsmentioned above. stiffequations as implemented in the subroutineIVPAG of the IMSL package. A. Bubble dynamics In the calculations to be described here we have used IV. RESULTS for the dimensionlessforcing soundamplitude the value The modeldescribed above contains a largenumber of e=0.93. In this choicewe have been guidedby a desireto parametersand an extensiveinvestigation of the relevant usethe lowestpossible level compatiblewith the presence parameterspace would require a major effort.Aside from of clear chemical effects. The reasons are twofold. In the its magnitude,a compellingreason not to embarkin such first place, although state-of-the-art,the mathematical a taskis suggestedby the imprecisionwith whichthe rates modelwe useis probablyinvalid at forcingsmuch above 1 used in the chemical kinetics calculations are known at the bar both in its chemical kinetics and bubble dynamics as- elevatedtemperatures and pressuresencountered in the pects.Second, with increasingpressure amplitudes, col- calculations to be described. In this situation, it is better to lapsesbecome sharper and sharper and smaller and smaller focuson the genericfeatures of the processwhich should time stepsmust be usedwith the consequenceof a consid- be relativelyrobust and independentof the specificnature erableincrease in computationtimes. The data of Barber of the systeminvestigated and of the uncertaintiesin the and Putterman ( 1991) and Barber et al. (1992) have been details of the kinetics. In this spirit, we shall therefore obtainedwith pressureamplitudes covering a range be- studyin detailthe specificcase of argonbubbles in water at tween somewhatless than one bar to over 5 [seeFig. 1(a) a frequencyof 21 kHz, which is closeto the frequency in the reference].Crum and Gaitan'slevels are between1.1 range of some recent experiments(Gaitan and Crum, and 1.5 bar. 1990; Gaitan, 1990). These experimentshave been con- We show in Fig. 1 the normalizedmaximum radius ductedwith air bubblesbut, as explainedbefore, a chemi- attainedby a bubblein steadyoscillations under a forcing
252 d. Acoust.Soc. Am., Vol. 94, No. 1, July 1993 Kamathet aL: A theoreticalstudy of sonoluminescence 252 24 10 •
22 I 12 20 10 • 110 18
16 1.08 10ø'.•. •106 12
10 1.04 10-' 08 06 I02 • I 00 04 • • I 0 -• 57 58 59 60 6 62
0 9810 1.•2 14• 16• 18• 20i 2•2 24• 26• 2.•8 30
FIG. 2. Normalized radius (solid line, left scale) and internal pressure (dotted line, fight scale,also shown in Fig. 3) for a 26-/•m argonbubble oscillatingat at 21 kHz and 0.93 relativepressure amplitude. Steady-state FIG. 4. Normalized temperaturedistribution in the liquid outside the conditionshave essentiallybeen reachedby the 10th cycle shown in the bubbleof the previoustwo figuresat the momentof peak centertemper- figure. ature (upper curve) and of maximum radius (lower curve). e=0.93 as a function of its equilibrium radius. The ambi- ering conductioneffects in the gas. Indeed, from Fig. 3, it ent pressureis 1 bar and the liquid temperature20 ø. At the can be seenthat the centertemperature keeps rising due to driving frequencyof 21 kHz usedin thesecalculations, the conduction as the bubble hovers around its maximum ra- resonant radius Rres is 150 ftm and therefore the region dius. For this example, at the point of maximum radius, comprisedin the figure is for 0.067
10z 3000 lO
9 10•i 2500 8 2000 7
Ch10 ø 1500 • 5
1000 4 10 • 3 500 2 10-z 517 518 5'9 •'0 61I 612, (• 1 0 O0 01 02 03 04 05 06 07 08 09 10 FIG. 3. Normalized internal pressure(dotted line, left scale,also shown y/R(t) in Fig. 2) and internal temperatureat the bubblecenter (solid line, fight scale) for a 26-/•m argon bubbleoscillating at 21 kHz and 0.93 relative FIG. 5. Normalizedtemperature distribution in the gasinside the bubble pressureamplitude. Steadystate conditionshave essentiallybeen reached of the previousthree figuresat the moment of peak center temperature by the 10th cycle shownin the figure. (upper curve) and of maximum radius (lower curve).
253 J. Acoust. Soc. Am., Vol. 94, No. 1, July 1993 Kamath et aL: A theoretical study of sonoluminescence 253 side and inside the bubble at the instant at which the center 10'0 , , , • , , 3000
10 9 reachesits maximum temperature(upper line) and at the 2500 instant of maximum radius (lower line). It can be seen -- OH 108 2000 from Fig. 4 that the temperatureincrease predicted to oc- HO• T(K) cur at the bubble wall is of the order of 40 K, so that the 10 ? 1500 • meantemperature gradient is about2.5X105 K/mm! 106 Clearly thesehighly unusualconditions can only be main- 1000 105 tained due to the extremely brief persistenceof the high 500 temperatureat the bubble center,which is of the order of 10' j/ • •- \\ j • 100 ns. 10s • I • • 0 57 58 59 60 ...... 61 ....62...... These resultsare typical and demonstratethe points wt made above on the smallnessof the surface temperature rise. They however appear, at least superficially,to be in FIG. 7. Center temperature(dash-and-dots, right scale) and radical contrast with results of Suslick's (1986, 1990) who has numberduring the 10th oscillationcycle of a 22.5-/•mbubble. This value producedconvincing experimental evidence that molecular of the radiuscorresponds to a peakin the responsecurve of Fig. 1. speciesinitially in the liquid phaseundergo reactions that only occur at temperaturesin excessof 2000 K. Suslick that extremely violent shapedistortions take place at the (1990) suggeststhat thesehigh temperaturesare reached pressurelevels used by Suslickwhile only mild onesoccur in a very thin liquid shell surroundingthe bubblesurface. at the lower intensities used by Gaitan, Putterman, and It should be stressedthat these experimentalresults have co-workers.From the experimentalevidence of stably os- beenobtained at pressureamplitudes of over 10 bars,far in cillating bubblespresented by theseauthors, one deduces excessof the onesused for the presentcalculations, so that that, in a parameterrange similar to the one we consider, a direct and quantitativecomparison with our findingsis any nonsphericaleffect, if present,is not strongenough to not possible.However, it would seemthat our conclusions causebubble break-up or "dancing." Hence, in spite of on the modestheating of the liquid surfacecast some doubt beingbased on the assumptionof sphericaloscillations, our on Suslick'sinterpretation. A more plausibleexplanation model may be a reasonableapproximation to the real phys- may be the following. It is well known that the spherical ical situation. shape of an oscillating bubble is unstable (Plesset and Accordingto the excitedOH hypothesis,the sonolu- Prosperetti,1977). The instabilitybecomes particularly vi- minescenceeffect depends on the total numberof hydroxyl olent during periodsof large accelerationsdirected into the radicalsproduced in the bubble.As the gastemperature is liquid such as occur near the end of the collapseand early not uniform, in principleit would be necessaryto calculate stagesof the rebound. Pulsating bubbleshave been ob- the OH concentration at a number of radial locations and served to shed other small bubblesprobably as a conse- determinethis number by integratingthe resultsof such quenceof this instability (Nyborg and Hughes, 1967). The calculationsover the radius. For simplicity, we have not reciprocal processof tiny droplets projected into the hot done this but we have only consideredthe temperature bubble core can also occur, and this appearsto be a more history at the bubble'scenter. In this way, strictly speak- plausiblemechanism by which moleculesinitially in the ing, we can only calculatethe concentrationsin a small liquid phaseare exposedto high temperatures.The severity fraction of the bubble volume near the center. If we were to of this surfacedisruption increases strongly with the inten- presentour resultsby plottingthese values versus time, we sity of the driving soundfield. It is thereforequite possible would not only be seeingthe effectof the chemicalkinetics but also those of the bubble expansionand contraction which do not have a direct bearingon the total numberof 10'0 [ • I I ' • 3000 moleculesand radicals actually present. Therefore, we ! 09 2500 -- OH .... II 106 I HO•0 2000 10 '0 , , , , , , 3000 I T(K) 10• • 10 9 II 1500• 2500
10• II , 108 2000
I 1o'L10•L / It• I ii• ., I I -1 -]5001000 1500• 106
10$11 I I I I I •0 1000 57 58 59 60 61 62 105 wt 500 10' FIG. 6. Center temperature(dash-and-dots, right scale) and numberof OH (solid), H (dots), O (medium dashes),and HO2 (long dashes)for 10 s 0 a 20-/•m bubbleat 21 kHz and 0.93 relative pressureamplitude. These numbersare obtainedby multiplyingthe concentrationevaluated in cor- respondenceof the centertemperature by the instantaneousvolume of the FIG. 8. Center temperature(dash-and-dots, right scale) and radical bubbleand thereforemay be in error by one or two ordersof magnitude. numberduring the 10th oscillationcycle of a 22.5-/•mbubble. This value See text. of the radiuscorresponds to a peak in the responsecurve of Fig. 1.
254 J. Acoust. Soc. Am., Vol. 94, No. 1, July 1993 Kamath et aL' A theoretical study of sonoluminescence 254 10 tø 3000 10 •ø ,
109 109 2500 3000
10 e 10 e 2500
2000 _ I 2000 10 ? 1500 1500•.. 106 106107 ......
1000 1000 105 105 10' 500 10' I 500
103 ,5'7 5• 59I 60 61i 6• 0
FIG. 9. Center temperature (dash-and-dots,right scale) and radical FIG. 11. Center temperature (dash-and-dots,right scale) and radical numberduring the 10thoscillation cycle of a 25-/zmbubble. This valueof numbersduring the 10th oscillationcycle of a 26-/zm bubble.This value the radius correspondsto a relative minimum in the responsecurve of of the radiuscorrespond to a peak in the responsecurve of Fig. 1. Fig. 1.
and compressionoccur is crucial, and this time scale is have decidedto show the computedconcentrations multi- determined by the radial dynamics. At a frequency far plied by the instantaneousbubble volume. In this way it is below the resonancefrequency of the bubble,compression easierto get a feelingfor the absolutenumbers of radicals and expansionfollow the driving pressureand the bubbleis and moleculeswhich are the quantities of interest, al- essentiallyisothermal. As dynamical effectsbecome more though the levelsindicated will representoverestimates of important, the bubble spendsmore and more time in the the actual numbers.On the basisof Fig. 5, the central expandedstate and collapsesare shorter and shorter. The region of the bubble where the temperatureis relatively hypothesesof the crude model (30) are thereforeincreas- uniform can be estimated to extend to about 25%-50% of ingly applicableand the maximum temperaturemust in- the radius, which correspondsto about 0.01 to 0.1 of the crease.This simpleargument shows that, over the spanof volume. A more realistic estimate of the actual molecular a few/zm in the value of the equilibriumradius, a threshold numberswould thereforebe obtainedby subtractingone or for radical production must be reached. two ordersof magnitudefrom the scalesused in the figures We can follow the consequencesof this gradual in- below. creasein the importanceof inertial effectson sonolumines- cencein Figs. 6-12. Figures6, 7, 9, and 11 show the center B. Argon bubbles temperature (dash-and-dottedlines) and the correspond- As the discussionthat followswill show,the gastem- ing radical numbersfor bubbleradii of 20, 22.5, 25, and 26 peratureis the dominantvariable in the productionof OH /zm. Figures 8, 10 and 12 show the numbers of several radicals.Our resultsshow that this quantity is not neces- molecularspecies for radii of 22.5, 25, and 26/zm. All these sarily correlatedwith the maximumradius reached during resultsare for the 10th cycle of oscillation.The solid line in the oscillations.Indeed, for example,the maximum tem- Figs. 6, 7, 9, and 11 is the number of the OH radicals. A peraturereached for the caseR0= 25/zm (the secondmin- very rapid increasetakes place when the temperaturerises imum in Fig. 1) is closeto 2300 K, while for 15/zm, for above approximately 1400 K. The temperature effect which the maximum radiusis larger, the maximum tem- alone, however,is not sufficientto explain all the features perature is only 1450 K. Evidently, since conductionef- of thesecurves. For instance,in Fig. 11, there is a surgeof fects are important, the time scaleover which expansion OH radicals for wt_•60.03, although the center tempera- ture is only slightly above 1000 K.
10 tø , , , , 3000
109 10•ø: • • • • • 3000 2500 10 s 10 e T(K) - 2500 j -- H20 2000 1oe I•)_ .... 10 ? II - - - It 2 -- HzO 2000 1500 / !/ ___Hz 106 • 107 I I/ ---- H202 15oo •;• 1000 105 I • 1ooo 10' 500 I ii _'i'-'-'"...... :': I/ - '- -' 10 s i i i ,i i i / 57 58 59 60 61 62 1010's 5•7 5•8 5•9 "160I• /Idl •\•/I •\•-'116• kx•5000 FIG. 10. Center temperature (dash-and-dots,right scale) and radical numbersduring the 10th oscillationcycle of a 25-/zmbubble. This value FIG. 12. Center temperature (dash-and-dots,right scale) and radical of the radiuscorresponds to a relativeminimum in the responsecurve of numbersduring the 10th oscillationcycle of a 26-•m bubble.This value Fig. 1. of the radiuscorrespond to a peak in the responsecurve of Fig. 1.
255 J. Acoust.Soc. Am., Vol. 94, No. 1, July 1993 Kamath et aL' A theoreticalstudy of sonoluminescence 255 0 150 3000 10 '0 3000 ß 0 125 ' 109 2500 0 100 ,, /_.,\ .... H 2500 / \ T(K) 10 s / \ Hz0 2000 / \ --- Oz 2000 o0 o•050 if•.l...."'.., ii \ --- H• / \ -- H•O• • 0025 / ', 1500• • 107 / \ 15oo -0 000 ,- '" --..---. ", • • 106 I / /• '" 1000 105 -0 050 I -R 500 10' /•...... •.•.&...... 500,ooo -0075 ,• .•RI• 0 10 s • •. - - - r - • • -•-• - - -0 100 5919659 98 60O0 6002 6004 6006 6010860 0 3 40 3 42 3 44 3.46 3 48 3 50 3 52 3 54 3 56 3 58 3.6• COt
FIG. 13. Contribution of reactions Nos. 3, 4, 7, 8, 15 of Table I to the FIG. 15. Expandeddetail of the previousfigure near the first collapse fight-hand sideof the OH kinetic equationaround the time of peak center time. temperature(dotted line) for a 26-/•m argonbubble.
13, Nos. 4 (dotted line) and 7 (medium dashes), are seen To understandthis behavior it is necessaryto consider to play only a minor role, and the remaining reactionsof in greater detail the chemical reaction rates. We plot in Table I are of yet smaller importance. Fig. 13 the contributionto the right-hand sideof the equa- Another interestingaspect is the transient by which tion d[OH]/dt .... given by reactionsNos. 3, 4, 7, 8, and the steady state describedis established.For R0= 26/•m 15 of Table I. The signconvention is suchthat the quantity we show in Fig. 14 the radical numbersduring the first plotted is positive if the reaction proceedsfrom left to cycle, to be compared with Fig. 11. In Fig. 15 a magnifi- right. Only a tiny and greatly enlargedfraction of the cycle cation of the portion of the previous figure around the around the time of maximum center temperature (dotted bubble collapsetime ot= 3.44 is presented,and the corre- line) is shown here. In dimensionalunits the correspond- spondingreaction rates are givenin Fig. 16. On the basisof ing duration is about 1.14 •s. It can be seenthat the ear- these results one can form the following picture of the liest OH production mechanism to become important is sequenceof events.As soon as the temperature and pres- reaction No. 15 (long dashes), i.e., the dissociationof sure becomesufficiently high, water moleculesstart disso- H202. This molecule, once produced, remains relatively ciating according to reaction No. 8, H+OH stableuntil it is destroyedby high-energycollisions and is +M,-H20 + M. When a sufficient amount of atomic thereforean important "memory mechanism"for the sys- hydrogenhas formed, reaction no. 3, OH + H 2,- H + H20, tem. As a consequenceof this dissociation,the H202 mol- startsgoing, additional OH is formed, while H decaysand eculesare rapidly depleted(cf. the long-dashedline in Fig. H 2 beginsto appear. The presenceof a significantamount 12). At about the point where this happens,reaction No. 3, of OH also opens the way to atomic oxygen formation OH + H 2= H + H20 (solid line) starts producing OH, accordingto reactionsNos. 4 and 7. soonjoined by No. 8, H + OH + M = H20 + M (dash-and- Sincethe other reactionsare found to play only a mi- dot line). In order to proceedfrom fight to left, as indi- nor role in the processesdescribed above, one may enquire cated by the negativesign of the correspondingrates, the whether a simplified model consistingonly of reactions first of these two reactionsneeds H radicals that Fig. 11 Nos. 3, 4, 7, and 8 would lead to similar results. We have showsto be presentin sizeablenumber. As a matter of fact, found that the answer is in the affirmative with differences in all the examplesshown, the number of H radicals re- in the number of OH radicals of the order of 10% for the mains essentiallyconstant throughout the cycle and in- examplespresented here. Qualitativelythe major difference creasesfrom about 103 for Ro--20pm to about107 for Ro--26 pm. The two other reaction rates plotted in Fig.
0 005 , • • 3000
0 004 10 '0 3000 2500 OH 0 003 0 .... H 0 002 2500 2000 T(K) 0 001 10 s H•0 2000 0• • 0 000 1500 --- H• 107 -0 001 t HzOz 15007•, 1000 106 -0 002 \ / --- T(K) '- -0.003 1000 5OO 105 -0 004 --- R15 10' 500 -0.00•.40 3.41 3.42 3.43 3 44 3 45 3 46 3 47 3 48 3 49 3 5 cot 10 s • 0 0 I 2 3 4 5 6 FIG. 16. Center temperature(dotted line) and contributionof reactions Nos. 3, 4, 7, 8, 15 of Table I to the fight-hand side of the OH kinetic FIG. 14. Radical numberduring the first cycleof oscillationof a 26-/•m equationaround the time of the first collapse(dotted line) for a 26-/•m argon bubble. argon bubble.
256 J. Acoust. Soc. Am., Vol. 94, No. 1, July 1993 Kamath et aL' A theoretical study of sonoluminescence 256 10 9 •
1 0 8 1 0 8 Ne
10 ? 10 ?
1 0 6 1 0 8
1 0 5 1 05
I 10• 57 518 519 10 • 517 518 519 61 62 •t
FIG. 17. Number of hydroxyl radicalsduring the 10th cycleof oscillation FIG. 18. Comparisonof the numberof hydroxylradicals during the 10th of a 26-/•mbubble driven at a 0.93 relativepressure amplitude (solid), of cycleof oscillationof a 26-/•m bubblecontaining argon (solid line) and a 26-/•mbubble driven at a 1 relativepressure amplitude (dash-and-dots), neon (dashedline). The thermal conductivityof neon is 3-4 timesthat of and of a 27-/•m bubbledriven at a relative pressureamplitude of 0.9 argon. (dashes).
biggerbubble at the lowestforcing is that, in that case,the is the absenceof HO2 production,which doesnot seemto correspondingresponse curve (similar to Fig. 1) is such have major consequencesanyway. On the other hand, it that R0= 26/•m doesnot correspondto a peakand the OH should be stressed that this result is limited to the small productionis thereforeartificially low. Comparisonof the parameterrange that we have investigatedand we do not three curves illustrates the tremendous sensitivity of the have sufficientelements to turn it into a generalconclusion. processto evensmall changes in the pressureamplitude, as first pointed out by Flynn (1964). C. Other effects We examine the effect of the thermal conductivity of the gas by comparingin Fig. 18 the argon results (solid Having looked in detail at the casesof argonbubbles, line) with thosefor neon (dashedline), both for a 26-/•m we now briefly explorethe sensitivityof the resultsto sev- bubble. The same chemical kinetic formulation can be used eral parametersand conditions. in both cases.The thermal conductivity of neon is about 3 In the first place, it is clear from the precedingdiscus- to 4 times that of argon between 300 and 800 K. Corre- sionthat, in the frameworkof the presentmodel, the liquid spondingly,the peak numberof hydroxylradicals is found temperaturehas a negligibleeffect on the radial dynamics. to be about a factor of 5 smaller. In consideringthe impli- Since,however, the OH radicalsultimately derive from the cationsof this comparisonit shouldbe kept in mind that, breakup of the water vapor molecules,one shouldfind a due to the difference in thermal conductivities, the internal proportionalitybetween the total numberof hydroxyl rad- pressureis differentin the two cases. icals and the water vapor pressure.We have tested this The results of chemical kinetics calculations for the prediction of the model and found it confirmedby the temperaturehistories at two other positionsin the bubble, numerical results.For example,at 60 øC, the vapor pres- r/R(t) •_0.464 and 0.585, correspondingto volume ratios sure is approximately10 times that at 20, and so is the of 10% and 20% respectively,are shown in Fig. 19. The calculated number of OH radicals. The sonoluminescent difference between the center (solid line) and the 20% emission should exhibit a similar increase and, since the vapor pressuredepends exponentially upon the liquid tem- perature, it should be possibleto take advantageof this 1 0 9 effectexperimentally. It shouldbe stressedthat this remark Tv/v ----' T ' TOT=10•. is only applicableall other conditionsbeing equal (in par- ,,pV/VToT=20% i i s'V/VToT=0% ticular, the dissolvedgas concentration and the soundpres- 1 0 8 sure amplitude), which may not be easyto achieveexper- imentally. Of course,as the liquid temperatureincreases, • 0? evaporation-condensationeffects at the bubble wall, ne- glectedin the presentmodel, becomemore and more im- portant (Prosperettiet al., 1988). However,for water, this shouldnot happenbelow about 60 øC. The resultspresented before were for a dimensionless driving pressuree=0.93. We comparein Fig. 17 the num- 519 610 61 ber of OH radicals during the tenth cycle for that case (solid line) with the numberfor a 26-/•m bubbledriven at FIG. 19. Comparisonbetw6en the numberof hydroxylradicals produced e= 1 (dash-and-dottedline) and a 27-/•m bubbledriven at at differentlocations inside a 26-/•m argon bubble driven at a relative e=0.9 (dashed line). The reason to consider a slightly pressureof 0.93.
257 J. Acoust. Soc. Am., Vol. 94, No. 1, July 1993 Kamath et aL: A theoretical study of sonoluminescence 257 result (dash-and-dottedline) is quite considerableas had higher,and so will be that of OH radicalsthat ultimately been anticipated. derive from the collisionaldissociation of the vapor mole- Finally, it may be noted that examplesof chaotic ra- cules. dial oscillationshave been reported in the literature at pres- On the basis of our numerical results the maximum sure amplitudes of the same order as the one used here number of OH radicals reaches every cycle the order of (Parliz et al., 1990). As shownin Kamath and Prosperetti 107-108.In orderto accountfor the temperatureeffect (1989), theseresults are in reality due to the neglectof the acrossthe bubble, one or two orders of magnitudeshould thermal dissipationaffecting the oscillations.Chaotic be- be subtracted from these numbers as mentioned above (see havior doesset in, but only at larger forcings,as is implic- Fig. 19). It is rather remarkablethat a recent experiment itly confirmed by the extreme regularity observedin the on sonoluminescence(with air, rather than argon,bubbles, recent experiments.In any event, our study suggeststhat a but in the same range of frequenciesand pressureampli- sonoluminescentpulse is emitted whenever a sufficiently tudes)estimates that about106 photons are emitted per large compressionand heatingof the bubblecontents takes pulse (Barber and Putterman, 1991). Of course,our cal- place. At low forcing, this would occur with the samepe- culation gives the number of radicals instantaneously riodicity as the soundfield. At higher forcing (provided of presentin the bubble,which doesnot necessarilycoincide coursethat the bubble can still be made to levitate stably), with the number of hydroxyl-radicalproducing events. An one might expect to find the same subharmonicmodula- estimate of this latter quantity may be obtained by com- tions of the light emissionthat are known to occurwith the paring the time scalefor productionwith the time during radius-timeresponse and, ultimately, a chaotic behavior. which the gastemperature remains elevated. If the former were found to be much shorter than the latter, one would D. Observability of sonoluminescence concludethat many "generations"of radicalsare produced (and recombine) during a single collapseevent. An esti- In practice, observabilityof stable sonoluminescence emissionby a single bubble requires a delicate balance. mateof the hydroxylproduction time scale rp maybe ob- tained from First of all, the soundamplitude and the gasconcentration in the liquid must be such that the bubble stabilizeswith [OH] respectto rectified diffusion.Such a stability point must occur for a value of the radius such that the rectified dif- r•,= 5;production, (31 ) fusion thresholdcrosses the descendingportion of one of where 5; productionstands for the all the OH-producing the resonancepeaks shown in Fig. 1. It is only in these terms appearing in the right-hand side of the OH rate conditionsthat the equilibrium can be stable.If the bubble equation.It is foundthat the productiontime scaleis of the radius increases slightly, the oscillation amplitude de- order of 1% of the soundperiod, and is thereforecompa- creases(see Fig. 1) and rectifieddiffusion is insufficientto rable with the duration of the elevated gas temperatures. prevent gasloss from the bubble.The radius thereforede- This suggeststhat at most a few "generations"of hydroxyl creases,and tends to return to the previous value. The radicalsare producedper collapseevent so that the com- converseprocess occurs if the bubble radius were to de- puted number of radicals instantaneouslypresent in the creaseslightly. bubblecoincides in order of magnitudewith the numberof A secondrequirement for the observabilityof the ef- hydroxyl-producingmolecular processesactually taking fect is that the sound intensity be sufficient to trap the place. bubble under the action of acoustic-radiation forces. Third, On the basis of these considerationsone can perhaps conditions must be such that a sufficient temperature is account for the observability of stable sonoluminescence reached inside the bubble upon compression.In view of with the naked eye. A reasonablenumber for the minimum theserequirements, observation of stablesonoluminescence photon count necessaryto form a perceivableimage is shouldbe easierwith a moderatelydegassed liquid. Indeed, around 100. At a distanceof about 20 cm the solid angle the lower the gas concentration,the higher the rectified subtendedbythe eye is of the order of 10-4 to 10-3rad, so diffusion threshold, so that equilibrium against diffusion thatthe bubble should emit a totalof 105to 106photons to can be achievedat acousticpressures sufficiently large to be observable. This coincides with our numerical results trap the bubble and induce strong collapses.On the other and indicatesthat the effectshould be directly visiblein the hand, a lower limit to the gas concentrationin the liquid parameterrange that we have studied. would in practice be set by the onset of shape instability The previousconsiderations depict a plausiblescenario and erratic motion of the bubble both of which become and are in qualitative agreementwith observation.How- stronger with increasing pressure amplitudes (see, e.g., ever, it shouldnot be forgottenthat, for an OH radical to Eller and Crum, 1970;Prosperetti, 1984; Horsburgh, 1991; give rise to a photon, it must first be excited to a state Benjamin and Ellis, 1990). At a given frequency, one abovethe groundlevel, a situationthat we have implicitly would therefore expect an optimal "window" of gas con- assumedto occur with high probability.Unfortunately, we centration. are unable to assesswhether this is a reasonableassump- A further parameterto which the occurrenceof stable tion. sonoluminescenceshould be sensitiveis the liquid temper- One may also speculateabout sonoluminescencein a ature. As noted before, all other things being equal, in a cavitatingfield, rather than for a singlestable bubble. If the warmer liquid the concentration of vapor molecules is sound intensity is very strong (say, peak pressureampli-
258 J. A½oust.So½. Am., Vol. 94, No. 1, July 1993 Kamath et aL: A theoretical study of sonoluminescence 258 tudesabove 4 bars or higher,depending on the frequency) the expandedstate. Calculationsfor neon, with a thermal the cavitation is vaporousand the lifetime of each bubble conductivity 3-4 times larger than argon, indicate a re- extendsto one or at most very few cyclesso that probably duced OH formation. only a small number of flashesare emitted by each bubble We have also calculatedthe temperatureof the bubble before it shatters.At lower sound amplitudes,however, surfaceand found that it changesat most by a few tens of rectifieddiffusion plays a very important role. Nuclei start K from the undisturbedliquid temperature.This finding growingby this processbut, as long as their radiusis much seemsto be in conflict with the reported fact that com- smaller than the resonanceradius, they pulsatefollowing poundspresent in the liquid, but not in the gas, phaseare the soundpressure with negligibledynamical effects. This exposedto very large temperatures (Suslick 1990). We situationcorresponds essentially to the left-handportion of have proposeda mechanismto resolvethe apparentpara- the responsecurve of Fig. 1. As the radius increases,the dox. seriesof peaks of Fig. 1 is describedin ascendingorder As can be seen,e.g., from Fig. 13, the persistenceof a until, at somepoint, the collapsesbecome strong enough to high temperature near the bubble center extends for causephotonic emissions. With further growth the bubble slightly lessthan 1% of the cycle,or approximately300 ns. responsedecreases beyond the resonancepeak to increase On thesegrounds it is therefore impossibleto accountfor again when the next one is encountered.[At the lower the exceedinglyshort duration (currently estimatedat 50 sound intensitiessome bubblesmay be "trapped" as de- ps) of the sonoluminescenceflash (Barber and Putterman, scribedabove for the single-bubblecase, but this should 1991; Barber et al., 1992). Recent work (Greenspanand becomemore and more unlikely as the pressureamplitude Nadim, 1993; Barber and Putterman, 1993; L/Sfstedtet al., increases as the bubbles would be much above the rectified 1993) suggeststhe formation of shock waves in the gas diffusionthreshold.] At somepoint a resonanceis encoun- duringthe compression cycle. A shockwith a speedof 104 tered for which the oscillationamplitude is too large for m/s would traverse 1/•m in 100 ps. Such a strong shock the bubbleto maintain its integrity. The bubble therefore could conceivablycause luminescence and ionization of the shatters producing microbubbles,which follow a similar gas molecules.In this case, the model describedin the history, possiblywith the addedcomplication of Bjerknes- present paper would very severelyunderestimate radical force-inducedcoalescence. At theselower amplitudes,the formation. bubblepopulation is in a stateof relativelyslow continuous Another possibilitymight be the following. The center evolution, with each bubble probably emitting for many of the bubble is the positionwhere radicals are first pro- cyclesduring severalstages of its life. duced. Shortly thereafter, however,additional radicalsare These considerations indicate that sonoluminescence generatedin the adjacent layers which could resonantly emissionsfrom stablesingle bubbles and from bubblefields absorb and scatter the radiation coming from the center may appearto behavequite differentlyexperimentally. For thus shutting off the observedlight. The effectivenessof example, it is found that bubble-fieldsonoluminescence in- resonantscattering in renderinggases effectively opaque is tensity decreaseswith increasingtemperature (Verral and well known since the classic 1904 experimentsof Wood. Sehgal, 1988). This result doesnot necessarilyfalsify our This mechanismmight be made more plausibleby observ- opposite prediction for the single-bubblecase as nucle- ing that, due to the condition OT/Or=O, the bubble core ation, and especiallyrectified diffusion--which is so impor- has a relatively homogeneoustemperature. However, in tant in the dynamicsof bubble fields that it would be hard the adjacentlayers, a temperaturegradient prevails and the to overestimateits significance--wouldalso be stronglyaf- coldestradicals would probablytend to be producedin the fected by temperaturechanges. ground rather than in an excitedstate. The time scale for this mechanismis dictated by the speedwith which the temperature"wave" movesoutward V. SUMMARY AND CONCLUSIONS from the bubble center. We have calculated with our model In earlier studiesof sonoluminescence,the tempera- that it takes slightly lessthan 1 ns for temperaturesin the ture of the gas in the bubbleshas always been estimated range 1500-2000 K to propagatefrom the center of the bubble to a distance of 0.1 times the radius. On this basis, rather crudelyand it has thereforebeen impossible to ob- tain reliable reaction rates. Here we have used instead ac- the size of the light-emitting region would be of the order of perhaps1% of the bubbleradius or less.A direct test of curate modelsfor the calculationboth of the temperature field and of the associatedchemical kinetics. In this way we this hypothesisdoes not appear straightforward.A conse- have obtainedreliable estimatesof the productionrates of quenceof the proposedmechanism would however be a hydroxyl radicalswhich are thought to be responsiblefor lengtheningof the duration of the light pulseas the sound intensity is lowered more and more toward threshold con- at least part of the light emitted by oscillatingbubbles. Heat transfer inside the bubble and across its interface ditions, as in this case the radical production would be limited to the very center of the bubble. hasbeen found to play a very importantrole in the process. In particular--and contrary to statementsfrequently en- ACKNOWLEDGMENTS countered in the literature--an estimate of the collapse temperature based on the adiabatic law may substantially The presentstudy has been supportedby NSF under underpredictthe peak gas temperaturebecause it ignores grant CTS-8918144. F. N. E.'s participationhas been sup- the heatingof the gasby conductionwhile the bubbleis in ported by AFOSR grant AFOSR-85-0147.
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