mie scattering used to determine spherical bubble oscillations

Mie scattering used to determine spherical bubbleoscillations

R. Glynn Holt and Lawrence A. Crum

Linearly polarized laser light is scattered from an oscillating, acoustically levitated bubble, and the scatteredintensity is measured with a suitable photodetector. The output photodetector current is converted into avoltage and digitized. For spherical bubbles, the scattered intensity Irel(ROt) as a function of radius R andangle 0 is calculated theoretically by solving theboundary value problem (Mie theory) for the water-bubbleinterface. The inverse transfer function R(I is obtained by integrating over the photodetector solid anglecentered at some constant 0. Using R(I) as a look-up table, the radius vs time [R(t)] response is calculatedfrom the measured intensity vs time [Iexp(R,t)I.

1. Introduction

Cavitation bubbles in liquids have been of interest toscientists since 1754, when Euler' first suggested thepossibility of cavitation in connection with his study ofturbine theory. With the recognition of cavitation asthe chief mechanism of erosion damage in ship propel-lers, turbines, pumps, and other hydraulic machineryin the early 1900s came an increase in both experimen-tal and theoretical studies of the phenomenon. Per-haps the first significant theoretical work dealing withthe dynamics of bubbles was that of Lord Rayleigh2 in1917 describing the collapse of a spherical cavity.

Beginning in the 1950s, the response of such bubblesto applied acoustic fields was the subject of muchtheoretical work. Papers by Hsieh,3 Plesset,4 Nol-tingk and Neppiras,5 ,6 Poritsky,7 and Hickling andPlesset8 greatly advanced the theory, resulting in adescription of the radius of a bubble as a function oftime involving a single, second-order nonlinear differ-ential equation. A more refined treatment of theproblem9 has led to still more complicated expressionsinvolving partial differential equations describing theinternal pressure and temperature of the bubble cou-pled to the radial equation of motion. However, themost important variable (on which most of the observ-able effects of cavitation depend) remains the radius asa function of time. There is a plethora of numerically

R. Glynn Holt is with Yale University, Department of MechanicalEngineering, P.O. Box 2159, Yale Station, New Haven, Connecticut06520. Lawrence A. Crum is with the National Center for PhysicalAcoustics, Coliseum Drive University M5 38677.

Received 21 August 1989.0003-6935/90/284182-10$02.00/0.

(© 1990 Optical Society of America.

generated radius vs time [R(t)] curves from differentmodels, but, until now, there has been no direct experi-mental verification of any of these predictions for sin-gle, stably oscillating bubbles.

There are a number of other groups whose interest inbubble dynamics is increasing. These include: shipwake researchers,' 0 who study the persistence of bub-bles in wakes and their passive and active contribu-tions to underwater sound propagation; ocean ambientnoise researchers,"1 who have recently presented evi-dence of the significant role of oscillating bubbles as anambient noise source; and biomedical researchers,'2"13

who are currently debating the possible side effects(both harmful and helpful) of acoustic cavitation in theuse of diagnostic and therapeutic ultrasound. Each ofthe phenomenon studied by these researchers dependsdirectly on the response of a bubble to an appliedpressure; i.e., the R(t) curve for the particular drivingpressure.

Thus, the motivation for this study was the need toproduce experimentally the radial response of a singlebubble to an applied acoustic field. This has beendone, and this paper describes the experimental meth-od employed to accomplish the task.

Section II contains the theoretical aspects of theproblem. Section II.A introduces (without deriva-tion) theoretical models for radial bubble oscillations.Numerical results are presented for comparison withthe experimental results presented in Sec. III. SectionII.B describes the theory of optical scattering from adielectric sphere for the relevant size parameter (ka)range, usually known as Mie scattering.'4 Numericalresults for the scattered intensity as a function of boththe scattering angle and the radius of the scatteringbubble are presented.

Section III contains the details of the experiment.Section III.A describes the general method used, inde-

4182 APPLIED OPTICS / Vol. 29, No. 28 / 1 October 1990

pendent of specific apparatus. Section III.B descrithe particular apparatus used in each phase ofexperiment. Section III.C describes the methodscalibrating the output current of (and hence the sctered light intensity incident on) the photodetecCalibration data are presented for the scattered liintensity. Section III.D describes the experimerprocedure followd to obtain I(t) curves and respo:curves as a function of equilibrium bubble radius.

Section IV contains selected results, and Sec. V discussion of those results, sources of error, andadvantages and limitations of the method used.

11. Theory

A. Bubble Oscillations

It may be argued that the ultimate aim of any thretical treatment of acoustic cavitation is the attament of an equation of motion describing the oscitions of the cavity volume, or, more specifically, 1motion of the gas-liquid interface. The general pmrlem is to find the pressure and velocity fields of 1two-phase medium. The starting point for any stundertaking must be the solution of the conservat:equations for mass, momentum, and energy, subjecisuitable boundary conditions at the bubble interfai

For the most general case when no symmetryinvoked, the problem can be almost intractableterms of obtaining any useful information about 1motion. An excellent treatment of this more geneproblem can be found in Hsieh's 1965 work,3 and interested reader is referred to his paper for mdetails.

The imposition of axial symmetry reduces the coplexity of the problem somewhat. Although this is Inext logical step, it is not typically the next steptheoretical studies since it allows all sorts of shEoscillations. What is normally done is to obtainequation of motion for a purely spherical interface, athen express the solution as a superposition of sphEcal harmonics. For details of the stability analysis,Refs. 15-17. If the driving pressure field is isotroover the length scale associated with the bubble, siface modes can only be excited parametrically, ahence should only appear as a threshold depend(quantity.

For driving pressures below this threshold, Ishape of the interface will remain spherical. One cthen obtain a single, second-order, nonlinear ordin,,differential equation describing the motion of the bible wall, with radius R as the dependent variable atime t as the independent variable. The crux of Iproblem then becomes that of determining inter]pressure p of the gas inside the bubble, which i:function of both R and t.

There are two ways of dealing with the inter]pressure. The first and most widely used5 6,18-24 melod is the polytropic relation,

p = Po( R ),

where p is the pressure of the gas in the bubble, R is theradius, K is the polytropic index, and the subscript zerorefers to equilibrium values. The index K can rangefrom 1 (isothermal oscillations) to the ratio of specificheats y (adiabatic oscillations).

There are a few problems associated with the use ofthe approximation in Eq. (1). First appropriate crite-ria governing the proper value of K exist only for smallamplitude, linear oscillations.25 26 Second, for nonlin-ear oscillations, it is shown in an ad hoc fashion27 that Kitself must vary nonlinearly. Third, the integral over acycle of p dv vanishes, resulting in no net energy lossassociated with the heating and cooling of the gas.9Since thermal damping has been shown to be the domi-nant form of dissipation for a wide range of equilibri-um radii and frequencies,2 it seems clear that a bettertreatment for the internal pressure must be devised.Attempts to incorporate an effective thermal dampingcoefficient into the viscosity term to remedy the situa-tion2 3 have met with only limited success.

The second method of modeling the internal pres-sure consists of applying the conservation laws to theinterior of the bubble. This has been done by Flynn,2 8

Hickling,29 and Prosperetti et al.9 in different fashions.For comparison with the radial results presented here,the formulation of Prosperetti et al. is used.

1. Radial OscillationsFor a detailed derivation and description of the

equations of motion, along with some numerical re-sults, the interested reader is referred to Ref. 9. Forthe motion of the bubble wall, Prosperetti et al. usedan equation of motion originally derived by Keller andKolodner 2 2 :

c ) 2 ( 3c)

= (1 +-)-I BW)- Pt +RC) P + dB(t)C PL [P C P~~~~c dt (2)

where R is the radius of the bubble, c is the speed ofsound in the liquid, PL is the density of the liquid, andps is the time-varying driving pressure. Dots denotetime derivatives. The liquid pressure outside bubblepB(t) is related to internal bubble pressure p(t) by

2ou 4=LRPMt = B(R,t) R R (3)

where AL is the viscosity of the liquid, and a is thesurface tension of the liquid.

To describe the pressure of the gas inside the bubbleas a function of time, one has

nlu,he P= [ (y -1)K -rial R "JR

a where -y is the ratio of specific heats of thesumed perfect) inside the bubble, K is the

rial conductivity of the gas, T is the temperature,Lh- expression is evaluated at r = R.

The conservation of energy is expressed byr -y-) t fr Or Y=1Y) Or Dp = V T,

(1 at +y'pR ay a a\yR -D - 2 ,

(4)

gas (as-thermaland the

(5)

1 October 1990 / Vol. 29, No. 28 / APPLIED OPTICS 4183

where the variable r is defined byrT

T = J K(O)dO, (6)

and a moving boundary is used by defining y = r/R(t).The Laplacian is taken with respect to the variable y,and the thermal diffusivity D for a perfect gas is givenby

D(p,T) = K(T) -1 K(T)T (7)C~p(p,T) j' p

Of the major assumptions used to derive these equa-tions,9 two seem especially important: (a) the pres-sure is spatially uniform inside the bubble, and (b)there is no mass diffusion across the bubble wall.

The first assumption requires the Mach number ofthe bubble wall (calculated with respect to the speed ofsound in the gas) to be small. Care must be takenwhen calculating violent collapse cases which occur insome subharmonic oscillations and transient events.This can become complicated, since the heating of thegas during the collapse changes the speed of sound inthe gas. The second assumption is only a problem ifone is interested in time scales much greater than thetypical oscillatory period of a bubble, such as would beneeded to study chaotic oscillations.30-33

For the purpose at hand, these assumptions will bevalid, since only driving pressures below the thresholdfor surface oscillations will be used, and the resultingmotions will be singly periodic with one or more har-monic components. Using a program written by Gai-tan34 following the methods outlined in Ref. 9, numeri-cal results have been obtained for comparison with thedata in Sec. III. As an overview of the expected re-sponse of bubbles driven at different pressures p,(t) =P sin(27rfdt) as a function of their equilibrium radius,Fig. 1 plots a family of theoretical resonance curves forP = 0.03, 0.04,... up to 0.24 atm, with a driving fre-quency fd of 24.4 kHz. In addition to the main reso-nance expected for linear oscillators, there are alsopeaks at RO/Rres = 0.5 and 0.33, corresponding to har-monic resonances with frequencies of 2fd and 3fd, re-spectively. Rres is defined in this case as the equilibri-um radius of a bubble whose linear resonancefrequency fo equals the driving frequency fd. Alsoapparent is a bending of the peaks toward lower valuesof RO/Rres, typical of a softening nonlinearity.

Equations (2), (4), and (5) are solved numerically inRef. 9, and some typical oscillatory solutions are pre-sented here for the particular numerical implementa-tion outlined in the appendix of Ref. 9. Recently,Kamath and Prosperetti introduced a more time-effi-cient and stable method for solving the system35 utiliz-ing a spectral technique. Initial results indicate thatthis method could be more useful for long-time inte-grations. In addition, Crum and Fowlkes recentlyincorporated rectified diffusion into Eqs. (2), (4), and(5),36 using a method proposed by Eller and Flynn.37

This method was also incorporated into Flynn's bub-ble model28 by Church.38 These developments shouldgreatly aid the attempts to model the (possibly chaot-

0.8

0.60

0E4

0.2

0.00.2 0.4 0.6 0.8 1.0

Ro/ R.e

Fig. 1. Theoretical resonance curves forP = 0.03,0.04,...,0.24 atmobtained by numerically integrating Eqs. (2), (4), and (5) with fd =

24.4 kHz.

ic) large amplitude oscillations for large driving pres-sures and long times.

B. Optical Scattering

Optical scattering of one form or another has beenthe only successful technique for obtaining experimen-tally the time-varying size of a bubble in a liquid. Inthe past, however, this approach has been limited tohigh speed photography,39

-40 holography, and holocin-ematography4 1-46 by Lauterborn et al., with no workdone on single periodically driven gas bubbles, butrather on transient cavities or bubble fields. An inter-esting light transmission experiment on a bubble fieldwas also performed, 4 7 eliciting, however, only the spec-tral characteristics of the motion.

Recently, Hansen48 used laser scattering on single,stably oscillating bubbles as a means of sizing suchbubbles. His work, along with some work by Marstonand Langley,49 50 suggested the possibility of using la-ser scattering to observe the time dependent behaviorof a bubble. This section presents the theoreticalaspects of the scattering of monochromatic, linearlypolarized plane lightwaves incident on a spherical gasbubble in a liquid. The numerical technique used tosolve for the scattering amplitudes and some numeri-cal results for relevant cases are also presented.

1. Mie TheoryThe problem at hand is determining the scattered

intensity as a function of spatial coordinates for thecase of plane, monochromatic, linearly polarizedlightwaves incident on a dielectric sphere. An excel-lent textbook treatment of the problem is given inKerker,5 ' Chaps. 3-5, and his notation is used in thissection. No derivation of the scattering amplitudes isgiven here. For a complete derivation beginning withMaxwell's equations, see Ref. 52.

If one defines the scattering plane ( = 0 plane) asthat plane containing the incident ( = 0) direction andthe direction of the scattered wave vector, then the


0

-2

-4

~--6

0 t

-8

-10

-120.0 25.8 51.5 77.2 102.9 128.6 154.3 180.0

0 (degrees)

Fig. 2. Theoretical scattered intensity as a function of scatteringangle (O is forward) for a size parameter of 661. This corresponds toa radius of 38.6 sum with respect to the argon-ion 488.0-nm line.

Only the parallel polarized component S2 is shown.

scattered intensity is given by

= 2 2 5212 COS2(, (8)

Is = 42 IS112 sin 20, (9)

where Io is the component of the scattered intensityparallel to the scattering plane, I, is the componentperpendicular to the scattering plane, r is the distancefrom the bubble center (r >> R), 0 is the azimuthalangle, and X is the wavelength of the light in the sur-rounding medium. The scattering amplitudes S1 andS2 are given in Ref. 51.

For comparison with the current experiment, theincident beam is now restricted to being polarized withthe electric field vector parallel to the X = 0 plane, thusrestricting interest to Io. Further, the relative intensi-ty Irel is defined as

47r2r210I S212 CS4rel - os22 (10)

The size parameter a = 27rm2R/Xo, where m2 is theindex of refraction of water (taken to be 1.33), R is thebubble radius, and Xo corresponds to the blue argon-ion line (488.0 nm).

2. CalculationsNumerical calculations to obtain Irel were carried

out using a modification of a program developed byWiscombe.53 Several checks of the numerical accura-cy were made against results found in Refs. 48-51 and54. All the theoretical results were obtained on a DECMicrovax II running in single precision FORTRAN.

Typical run times ranged from 30 min for an Irel vs 0curve to 15 h for a solid angle integration calculating Sf IreIdOdcb vs R.

Figure 2 is a plot of Irel vs 0 for 0-180° for a sizeparameter of 661, corresponding to a bubble of equilib-

x10512r

10

8

86

4

2

0 20 40 60 80 100 120R (microns)

Fig. 3. Theoretical scattered intensity as a function of the radius ofthe spherical scatterer (bubble). The wavelength in the bubble is488.0 nm, and the scattering angle is 80° from the forward. S2 isplotted corresponding to polarization of the incident beam parallel

to the scattering plane.

rium radius Ro = 38.6 Am for Xo = 488.0 nm. It is notimmediately obvious from this graph where the opti-mal angular location, in terms of maximum intensityand minimum diffraction structure for a photodetec-tor, should lie, if indeed there is an optimal choice. Itis at least clear that one would like to be at some 0 <900, since the intensity drops an order of magnitudefrom 70 to 1000.

For a given photodetector combination, the finitesolid angle subtended by the apparatus must be takeninto account by an integration over the appropriatelimits in 0 and 0. In general, this has the effect ofeliminating the fine structure and mitigating thecoarse structure, depending on the size of the angle.Based on a suggestion by Marston, 5 5 and following theexample of Hansen,52 calculations and measurementswere made at 0 = 800, near the critical angle.54 Otherdetectors and scattering angles were used for variousreasons, but these results are not presented here (seeRef. 56). Figure 3 shows Irei vs R for 0 = 80°, with X0 =488.0 nm. Ignoring fine structure, the intensity risesregularly with increasing radius, and R(I) is singlevalued.

Ill. Experiment

A. Method

Although a variety of detection schemes have beenemployed, the method has remained constantthroughout the course of the experiments. Figure 4 isa generalized schematic of the apparatus used in theexperimental measurements and serves as a guide forthe discussion throughout this section.

An acoustic levitation technique 5 7 was used to ob-tain a single, stably oscillating air bubble in watercontained in a resonating cell. Only bubbles with Ro <Rres could be obtained by this technique. In practice,


Fig. 4. Generalized schematic of the experimental apparatus.Shading indicates a variable component of the system.

this meant that bubbles ranging from -20 to 100 ,umcould be obtained.

Once a suitable bubble was obtained, a linearly po-larized laser beam was scattered from the bubble,whose position in the beam was maintained by pres-sure adjustment and micropositioner stages on whichthe levitation cell was mounted. For purely radialbubble motion, the scattered intensity I(t) at someangle from the forward is related to the radius R(t) in anonlinear fashion for which a transfer function I(R) iscalculable from Mie theory. For other symmetric mo-tions (shape oscillations), I(t) is related to the motionof the interface in some more complicated fashion;nevertheless, suitable methods of analysis of I(t)should elicit information about the motion. This sub-ject is only touched on briefly here and is discussedelsewhere (Ref. 58).

The scattered intensity I(t) was measured using adetection scheme described in the next section. Thedetection scheme converts the incident light intensityto a photocurrent ip(t) which is linearly proportional tothe input intensity Iexp(t), and this current is convert-ed to a voltage which is recorded for later analysis andgraphic output. The next section describes in detailthe apparatus used in the experiment.

B. Apparatus

1. Base EquipmentThe unshaded components in Fig. 4 comprise the

basic apparatus needed to levitate a bubble and ana-lyze the data. This section describes these basic com-ponents.

In the center is the levitation cell. It consists of two3-in. diam cylindrical piezoceramic (PZT-4) transduc-ers, poled to be driven primarily in the thickness mode.The transducers are joined concentrically by a 3-in.long glass tube. Watertight coupling is ensured by theuse of a silicone glue. A Plexiglas disk is glued to thebottom transducer, and the resulting container is filled

to a specified level with distilled, filtered (1.0-yum par-ticulate filter) water.

The cell is mounted on a three degrees of freedomtranslation stage, providing 0.001-in. resolution and 1-in. travel in each direction with the use of manualmicrometer drives. This arrangement allowed the po-sitioning of the bubble anywhere in the plane perpen-dicular to the laser's axis of propagation. In addition,a 3600 rotation stage with 0.01° resolution was mount-ed independent of but concentric with the cell. Thisstage provided a variable, calibrated mount for thephotodetectors. The entire apparatus, including thelaser head and necessary optics, was mounted on a 4- X6-ft optical table with self-leveling pneumatic sup-ports for vibration isolation.

A frequency synthesizer/function generator coupledinto a 75-W power amplifier was used to drive thelevitation cell with a periodic, sinusoidally varyingvoltage. The driving frequency fd was determined bythe desired resonant mode of oscillation of the cell.59

For all the calibrated experiments, an (r,0,z) mode of(1,0,1) was used, with fd = 24.4 kHz. The amplitudeand frequency of the function generator were manuallymodulated.

Once the data were collected and stored in somebuffer memory, they were transferred to a DEC VAX-station II for analysis and plotting usingGGPLUS-V3. 6 0 Completed graphs were stored on adisk as postscript files and output to a laser printer.

2. Detection SchemeThe detection scheme which proved easiest to cali-

brate used a 3-W argon-ion laser operating at 488.0 nmwith a single line power of 440 mW in the TEMOO modeas the scattering source. A rotating polarizer mountedon the laser ensured a 1200:1 linear polarization ratio,and the beam had a nominal li/e2 width of 1.0 mm.The polarizer was rotated so that the electric fieldvector was parallel to the scattering plane defined by 0= 0, which was also chosen parallel to the surface of theoptical table.

The detector used was an Oriel model 7080-1 photo-diode/preamp module with an integral optical trans-mission filter for the 488.0-nm line (Oriel model 52630)to reduce random light noise. The semiconductor ac-tive surface was very large (100 mm2), and for thisreason a lens was not needed to get adequate lightinput. The trade-off was a rather large rise time (1100ns) which limited the bandwidth to 1 MHz. Thephotodiode was operated in the photoconductivemode,6 ' with the resulting photocurrent ip input to theinverting input of the integral preamp. A feedbackresistor Rf = 100 k connected the preamp output tothe inverting input, and thus the output voltage of thepreamp was Vo(t) = ip(t)Rf. This voltage was fed intothe 1 Mg dc input of the LeCroy 9400 for observation,averaging (to improve the signal-noise ratio), and tem-porary storage before being transferred to the VAX-station II. The photodiode output was dc coupled tothe digital oscilloscope to enable the observation of thedc output voltage due to the constant light intensity


x104

15.0

12.5

10.0

7.5

5.0

2.5

0.0AA 0 20 40 60 80 100 120

R (microns)

Fig. 5. Results of integrating the theoretical scattered intensityover the solid angle subtended by the photodiode, for which thespread in the scattering angle was +4.81'. The wavelength in the

bubble was 488.0 nm, and the center angle was 800.

associated with the time-averaged scatter from thebubble. For linear oscillations, this time-averagedscattered intensity (and hence the dc voltage compo-nent) corresponds to the bubble's equilibrium radiusRo; i.e., (Iexp(Rt)) = Iexp(Ro).

The photodetector was placed at a center angle of800. Figure 3 shows IreI vs R for 0 = 80°, Xo = 488.0 nm.Only the S2 component is shown, corresponding topolarization of the incident electric field vector paral-lel to the scattering plane. Figure 5 shows the resultsof integrating over the solid angle subtended by thephotodetector, Oacc = 4.810 . The integration wasperformed by Gaussian quadrature after subdividingthe 9.620 0 interval into 500 partitions and calculatingIrel(R) for each partition. The 0 integral was evaluat-ed directly. The functional form is approximately R2,as one would expect from physical optics approxima-tions.4 9

C. Calibration of Photodetector Current

The electric signal to be measured is Vexp(Rt) =Gip(R,t)RL, where G is the total gain factor of theintermediate electronics, RL is the load or feedbackresistance, and ip is the photocurrent. The photocur-rent ip(R,t) = ARIexp(Rt), where S is the total respon-sivity of the photodetector in amps/watt, and A is thearea of the photosensitive surface. Finally, Iexp(R,t) =10 r r Irei[R(t)]ddq5, where the limits on the integralare determined by the particular photodetectionscheme used. Thus, Vexp(Rt) = 4) r Ir[R(t)]d0d,where 4) = GRLARIO is an apparatus-dependent con-stant to be determined empirically.

First, a stably oscillating bubble is obtained in thecell and moved into the beam. The output voltageVexp(R,t) is monitored on an oscilloscope. The timeaverage Vexp(Ro) = (Vexp(R,t)) is recorded. Only lin-early oscillating bubbles are used, since Ro = (R(t))only for linear oscillations, and hence (Iexp(Rt)) =Iexp(Ro) only for linear oscillations. As Vexp(Ro) is

x104

15.0

12.5

~BCD

=:

10.0

7.5

20 O1~ I I I I I

* o 20 40 60 80 100 120R (microns)

Fig. 6. Data (symbols) and theory (solid line) for the photodiodewith solid angle of +4.81o centered at 800. Wavelength in thebubble is 488.0 nm. The radii were obtained by a rise time tech-

nique.

being recorded, an independent measurement of R ismade using a rise time technique similar to that usedby Crum26 and Hansen.48 In this fashion a set ofcalibration data points is accumulated spanning thewidest possible range of equilibrium radii. Data werealso taken for a range of driving pressures to ensurethere were no spurious effects due to rf noise. Thecalibration constant 4b was then determined by takingthe unweighted average over all data points (and hencea wide range of Ro) of the ratio of the experimentalvoltage to the relative intensity, i.e.,

I'ex Vexp(Ro)

J J Ire,(R=Ro)ddp

N

'(%xp

1 =N

(lla)

(llb)

where N is the number of data points. Dividing thephotodiode output Vexp(R,t) by 4) gives the experimen-tal relative intensity, which can then be used to findthe radius R(t).

Dividing the calibration data Vexp(RO) by 4) and plot-ting these data on the same graph as the theoreticalintegrated relative intensity rr Irei(R)dOdk gives anindication of the accuracy of the calibration. Figure 6shows the calibration data vs radius for the Oriel pho-todiode. These data were taken over a period of twoweeks, with a temperature variance of ± 10 C maximumbetween any two runs. The solid curve is the integrat-ed relative intensity obtained in Fig. 5. The increasedscatter for large bubble sizes is attributable to twofactors. First, the rise time measurements becomemore difficult to make with larger bubbles due to thespeed of their ascent. Thus the spread in the value ofR becomes greater. Second, larger bubbles are harderto center in the laser beam: since the lie2 beamwidth


was only 1 mm, small positioning errors for the largerbubbles could easily have caused a variance in thedetected scattered intensity. A maximum estimatefor the error in a radius value determined using thiscalibration is -9% for large bubbles (R > 80 Am), and-4% for small (R < 40 gm) bubbles.

D. Procedure

1. Introduction

This section briefly describes the procedures used toobtain the data presented in Sec. IV. In all cases,distilled, filtered (1.0 Am) water was introduced to thecell. A preset height of the water in the cell wasmaintained, determining the resonance frequency fd.An initial bubble was obtained by increasing the pres-sure amplitude until cavitation occurred, and decreas-ing the pressure until the bubble was stably levitated.The cell was then tuned, i.e., the driving frequency wasswept while monitoring the bubble's levitation posi-tion to find the resonance frequency. This was deter-mined by minimizing the bubble's position in the z-direction, thus maximizing the acoustic force on thebubble, which is directly proportional to the pres-sure.57 In this fashion, the resonance frequency wasmaintained to within +0.1 kHz for all the data sets.

Using the micropositioners, the levitation cell wasmoved relative to the laser beam until the bubble wascentered in the beam. This position was determinedby monitoring the maximum dc component ofVexp(Rt) on an oscilloscope. With the bubble in thebeam center, the system was ready for data acquisi-tion.

2. R(t) CurvesOnce a record length was established, the bubble was

observed until an event or region of interest was at-tained. The digitizers were then manually triggered,typically (although not always) sampling at a rate of 1Msample/s. Voltages were continuously averaged touncover the signal buried in the random electricalnoise, then the averaged signal was stored.

The stored Vexp(Rt) data were input into a programwhich divided the voltage by ) to obtain Iexp(Rt).The exp values were then compared with the contentsof a 2-D array containing the pairs [ f Irel(R)d0dbR].If Iexp(ti) matched a tabulated value, R(ti) was written.If not, linear interpolation was performed to find asuitable R(ti). Analysis and plotting, discussed in Sec.IV, were then performed.

3. Response CurvesFor the response data, once a suitable bubble was in

the beam, three data values were recorded: Vmax, Vmin,and either Vave or t, the rise time, for determining Ro.Ro was varied in one of two ways. The first, applied ifthe driving pressure was low and the gas concentrationin the liquid was undersaturated, was to start with alarge bubble and let it dissolve, taking data as it dis-solved. It took -1-2 h to obtain a range of data fromRo 80 to Ro 30 gim. The second, applied if the

0.25

~-- 0.15

' 0.10

0000 ~ 00

0.05

0.00~ ~ 0

0.2 0.3 0.4 0.5 0.6 0.7 0.8Ro/ R.e

Fig. 7. Data (open diamonds, open squares, and filled squares) andtheory (solid lines) for, in respective ascending order, driving pres-

sures of 0.14, 0.20, and 0.24 atm with fd = 24.4 kHz.

driving pressure exceeded the threshold for rectifieddiffusion,6263 was to start with a small bubble and let itgrow toward resonance size. This took from -30 minto 1 h. The three voltages were then converted to radiiusing the look-up table described above, and a re-sponse measure (Rmax - Rmin)/2RO was plotted againstRo/Rres,

IV. Results

A. Radial Motion

1. Response CurvesTo compare with Fig. 1, Fig. 7 plots a family of

resonance curves for three different driving pressures.The discrete points are experimental data, and thesolid curves are numerical results generated using Eqs.(2), (4), and (5) for best-fit pressures of 0.14, 0.2, and0.24 atm, in ascending order. The data, though sparse,show good agreement with the theory. The salientfeature is the presence of the second harmonic reso-nance (corresponding to a response with a frequencycomponent of 2fd), which has been seen indirectly byCrum and Prosperetti,26 2762 but never before mea-sured.

2. R(t) AnalysisFigure 8 shows the temporal behavior of a bubble

with equilibrium radius Ro = 64.2 m, oscillating nearthe peak of the second harmonic resonance RO/Rres =0.5. Figure 8(a) plots the driving pressure ps, Fig.8(b)plots the voltage output Vexp of the photodiode, andFig. 8(c) is the radius R of the bubble obtained bothexperimentally and numerically [by integrating Eqs.(2)-(5)]. It should be noted here that the experimen-tal data were subjected to continuous averaging as theywere taken. The averaging lasted for fifty sweeps ofthe oscilloscope trace, corresponding to -1000 cyclesof the driving pressure. This was necessary to extractthe periodic signal (with more than one frequencycomponent) from the noise, which was essentially ran-dom in phase, with a broadband peak -50 kHz. Also,


0.25

0.00

-0.25 0.055 .

(b) 0.050

0.045

0.040

0.03576.00 ,

71.50 - Experimental - Theoretical (c)

67.00

62.50

200 400 600 800 100

Time (x 200 nsec)

Fig. 8. Nonlinear harmonic oscillation of a bubble of equilibrium

radius 64.2 jm. (a) Driving pressure vs time, fd = 24.2 kHz. (b)

Voltage output of the photodiode vs time. (c) Experimental andtheoretical bubble radius vs time.

a constant phase shift was added to the drive signal inFig. 8(a) to reflect the phase difference of the actualcell pressure and thus give the correct phase relation-ship between driving pressure and bubble response.

Although the maximum oscillation amplitude is pre-dicted very well, the magnitude of the second harmon-ic component in the theoretical curve is noticeablysmaller than the experimental value. This is essen-tially the same problem as that previously encounteredfor the second harmonic response in comparing theresults of an earlier theory with data taken by Crumand Prosperetti.6 4 The theory overestimated the val-ue of the nonlinear damping in the second harmonicregion. After comparing several data sets with theirtheoretical counterparts, it seems apparent that thecurrent theory, although better accounting for thethermodynamics of the bubble interior, also overesti-mates the nonlinear damping. Certainly, more data ofthis type need to be taken to quantify the discrepancy.

B. Nonradial Motion

Finally, several scattered intensity traces were re-corded similar to Fig. 9, where the experimental scat-tered intensity Iexp is plotted as a function of time in (a)and (b) is the power spectrum. This behavior is thesubject of a forthcoming paper, 5 8 where the motion isshown to be no longer spherical, but to exhibit thepresence of well-defined shape oscillation modes. Thecomplicated nature of the power spectrum indicatesthat quasiperiodic and/or chaotic motion is takingplace, and other methods of analysis will be applied toexplain the temporal behavior of the scattered intensi-ty.

V. Conclusion

A viable method for obtaining the temporal behaviorof oscillating bubbles has been presented and favor-

P.

0

~-10~L~

-20

, -30

-400 1 2 3 4

f / fd

Fig. 9. Response of a bubble undergoing shape oscillations. The

bubble equilibrium radius is -90 ,um, the driving pressure is -0.2atm, and the driving frequency fd is 24.4 kHz. (a) Voltage output of

the photodiode and (b) Power spectrum.

ably compared with the current theoretical results.65

While the data are not accurate enough to distinguishbetween different bubble models at low driving ampli-tudes, they were accurate enough to expose a systemat-ic error (the overestimation of the nonlinear damping)present in one of the models.

Sources of error are numerous, and no attempt willbe made here to enumerate all of them. Some of themore important ones are:

The lack of an experimental determination of thedriving pressure.

The rise time technique for obtaining an indepen-dent measure of the equilibrium radius. This gave riseto as much as 5% error in Ro. However, the fact thatthe peak experimental response occurred close to thepredicted second harmonic response lent much cre-dence to the method.

The deviation from plane wave incident due to fo-cusing of the laser beam by the cylindrical glass inter-face. This effect was ignored, since the distance fromthe bubble to the focal point was much larger than R0.66

The gain-bandwidth product limitation of the de-tector. Obtaining an effective bandwidth of 100 kHzconstrained the signal-noise ratio to a value of <1 forsmall amplitude, small bubble oscillations. Continu-ous averaging helped for periodic oscillations, but theassumptions about the long time scale for the effects ofmass diffusion were no longer as valid. However, forlarge amplitude, nonlinear oscillations, averaging wasnot necessary, and the signal-noise ratio was >>1.

This work has been supported by the Office of NavalResearch and the National Center for Physical Acous-tics. The author is indebted to his advisor, L. A.Crum, and to all the members of the acoustic cavita-tion group (past and present) for discussions, sugges-tions, and criticism. Thanks also to G. Hansen, P.


Marston, and A. Prosperetti for valuable discussions.The reviewers are especially to be thanked for valuablesuggestions and corrections. Finally, thanks to W.Wiscombe for sending his Mie scattering codes. Theexperimental work and analysis were performed at theNational Center for Physical Acoustics at Oxford, MS,in the Cavitation Laboratories of L. A. Crum.

References1. L. Euler, "Classe de Philosophie experimentale," Histoire de

l'Academie Royale des Sciences et Belles Lettres, Mem. R. 10,1754 (Berlin, 1756). pp 227-295; the remarks on the rupture ofthe liquid from the walls are made in Chap. 81, pp. 266-267.

2. Lord Rayleigh, "On the Pressure Developed in a Liquid Duringthe Collapse of a Spherical Cavity," Philos. Mag. 34, 94 (1917).

3. D-Y Hsieh, "Some Analytical Aspects of Bubble Dynamics," J.Basic Eng. 87D, 991 (1965).

4. M. S. Plesset, "The Dynamics of Cavitation Bubbles," J. Appl.Mech. 16, 277 (1949).

5. B. E. Noltingk and E. A. Neppiras, "Cavitation Produced byUltrasonics," Proc. Phys. Soc. London, Sect. B 63, 674 (1950).

6. B. E. Noltingk and E. A. Neppiras, "Cavitation Produced byUltrasonics: Theoretical Conditions for Onset of Cavitation,"Proc. Phys. Soc. London, Sect. B 64, 1032 (1951).

7. H. Poritsky, "The Collapse of Growth of a Spherical Bubble orCavity in a Viscous Liquid," in Proceedings, First U.S. NationalCongress on Applied Mechanics (ASME, New York, 1952), p.813.

8. R. Hickling and M. S. Plesset, "Collapse and Rebound of aSpherical Bubble in Water," Phys. Fluids 7, 7 (1964).

9. A. Prosperetti, L. A. Crum, and K. W. Commander, "NonlinearBubble Dynamics," J. Acoust. Soc. Am. 83, 502 (1988).

10. S. A. Thorpe and P. N. Humphries, "Bubbles and BreakingWaves," Nature (London) 283, 463 (1980).

11. A. Prosperetti, "Bubble Dynamics in Oceanic Ambient Noise"in Sea Surface Sound, B. R. Kerman, ed., 151 (Kluwer, Dor-drecht, 1988).

12. L. A. Crum, S. Daniels, M. Dyson, G. R. ter Haar, and A. J.Walton, "Acoustic Cavitation and Medical Ultrasound," Proc.Inst. Acoust. (U.K.) 8, 137 (1986).

13. R. E. Apfel, "Acoustic Cavitation: A Possible Consequence ofBiomedical Uses of Ultrasound," Br. J. Cancer Suppl. V 45, 140(1982).

14. G. Mie, "Beitrdge fur optik truber medien, speziell kolloidalermetall6sungen," Ann. Phys. (Leipzig) 25, 377 (1908).

15. M. S. Plesset, "On the Stability of Fluid Flows with SphericalSymmetry," J. Appl. Phys. 25, 96 (1954).

16. A. Prosperetti, "Viscous Effects on Perturbed Spherical Flows,"Q. Appl. Math. 35, 339 (1977).

17. A. Prosperetti and G. Seminara, "Linear Stability of a Growingor Collapsing Bubble in a Slightly Viscous Liquid," Phys. Fluids21, 1465 (1978).

18. H. G. Flynn, "Physics of Acoustic Cavitation in Liquids," inPhysical Acoustics, W. P. Mason, Ed. (Academic, New York,1964).

19. R. E. Apfel, "Acoustic Cavitation Prediction," J. Acoust. Soc.Am. 69, 1624 (1981).

20. W. Lauterborn, "Numerical Investigation of Nonlinear Oscilla-tions of Gas Bubbles in Liquids," J. Acoust. Soc. Am. 59, 283(1976).

21. Y. A. Akulichev, "Pulsations of Cavitation Bubbles in the Fieldof an Ultrasonic Wave," Sov. Phys. Acoust. 13, 149 (1967).

22. J. B. Keller and I. Kolodner, "Damping at Underwater Explo-sion Bubble Oscillations," J. Appl. Phys. 27, 1152 (1956).

23. A. Prosperetti, "Nonlinear Oscillations of Gas Bubbles in Liq-uids: Steady-State Solutions," J. Acoust. Soc. Am. 56, 878(1974).

24. J. B. Keller and M. Miksis, "Bubble Oscillations of Large Ampli-tude," J. Acoust. Soc. Am. 68, 628 (1980).

25. A. Prosperetii, "Thermal Effects and Damping Mechanisms inthe Forced Radial Oscillations of Gas Bubbles in Liquids," J.Acoust. Soc. Am. 61, 17 (1977).

26. L. A. Crum, "The Polytropic Exponent of Gas Contained WithinAir Bubbles Pulsating in a Liquid," J. Acoust. Soc. Am. 73, 116(1983).

27. L. A. Crum and A. Prosperetii, "Nonnlinear Oscillations of GasBubbles in Liquids: an Interpretation of Some ExperimentalResults," J. Acoust. Soc. Am. 73, 121 (1983).

28. H. G. Flynn, "Cavitation Dynamics. I. A Mathematical For-mulation," J. Acoust. Soc. Am. 57, 1379 (1975).

29. R. Hickling, "Effects of Thermal Conduction in Sonolumines-cence," J. Acoust. Soc. Am. 35, 967 (1963).

30. J. M. T. Thompson and H. B. Stewart, Nonlinear Dynamics andChaos (Wiley, New York, 1986).

31. W. Lauterborn, "Cavitation Bubble Dynamics-New Tools foran Intricate Problem," Appl. Sci. Res. 38, 165 (1982).

32. W. Lauterborn and E. Cramer, "Subharmonic Route to ChaosObserved in Acoustics," Phys. Rev. Lett. 47, 1445 (1984).

33. W. Lauterborn and E. Suchla, "Bifurcation Superstructure in aModel of Acoustic Turbulence," Phys. Rev. Lett. 53, 2304(1984).

34. D. F. Gaitan, NCPA University, M5 38677; private communica-tion.

35. V. Kamath and A. Prosperetti, "Numerical Integration Methodsin Gas-Bubble Dynamics," J. Acoust. Soc. Am. 85, 1538 (1989).

36. L. A. Crum and J. B. Fowlkes, "Acoustic Cavitation Generatedby Microsecond Pulses of Ultrasound," Nature (London) 319,52(1986).

37. A. Eller and H. G. Flynn, "Rectified Diffusion During NonlinearPulsations of Cavitation Bubbles," J. Acoust. Soc. Am. 37, 493(1965).

38. C. C. Church, "Prediction of Rectified Diffusion During Nonlin-ear Bubble Pulsations at Biomedical Frequencies," J. Acoust.Soc. Am. 83, 2210 (1988).

39. W. Lauterborn, "Kavitation durch Laserlicht," Acustica 31, 51(1974).

40. W. Lauterborn and K. J. Ebeling, "High Speed Holography ofLaser-Induced Breakdown in Liquids," Appl. Phys. Lett. 31,663(1977).

41. G. Haussmann, and W. Lauterborn, "Determination of Size andPosition of Fast Moving Gas Bubbles in Liquids by Digital 3-DImage Processing of Hologram Reconstructions," Appl. Opt. 19,3529 (1980).

42. W. Lauterborn and A. Vogel, "Modern Optical Techniques inFluid Mechanics," Ann. Rev. Fluid Mech. 16, 223 (1984).

43. W. Hentschel and W. Lauterborn, "High Speed HolographicMovie Camera," Opt. Eng. 24, 687 (1985).

44. W. Hentschel, H. Zarschizky, and W. Lauterborn, "Recordingand Automatical Analysis of Pulsed Off-Axis Holograms forDetermination of Cavitation Nuclei Size Spectra," Opt. Com-mun. 53, 69 (1985).

45. W. Lauterborn and A. Koch, "Holographic Observation of Peri-od Doubled and Chaotic Bubble Oscillations in Acoustic Cavita-tion," Phys. Rev. A 35, 1974 (1987).

46. A. Vogel and W. Lauterborn, "Time-Resolved Particle ImageVelocimetry Used in the Investigation of Cavitation BubbleDynamics," Appl. Opt. 27, 1869 (1988).

47. W. Lauterborn, J. Holzfuss, and A. Koch, "Recent Advances inCavitation Research at Gbttingen University," in Proceedings,1986 International Symposium on Cavitation, H. Murai, Ed.(Sendai, Japan, 1986).


48. G. M. Hansen, "Mie Scattering as a Technique for the Sizing ofAir Bubbles," Appl. Opt. 24, 3214 (1985).

49. P. L. Marston, "Critical Angle Scattering by a Bubble: Physi-cal-Optics Approximation and Observations," J. Opt. Soc. Am.69, 1205 (1979).

50. D. S. Langley and P. L. Marston, "Critical-Angle Scattering ofLaser Light from Bubbles in Water: Measurements, Models,and Application to Sizing of Bubbles," Appl. Opt. 23, 1044(1984).

51. M. Kerker, The Scattering of Light and Other ElectromagneticRadiation (Academic, New York, 1969).

52. G. M. Hansen, "Mie Scattering as a Technique for the Sizing ofAir Bubbles," Ph.D. Dissertation, U. Mississippi, Oxford (1983).

53. W. J. Wiscombe, "Improved Mie Scattering Algorithms," Appl.Opt. 19, 1505 (1980).

54. D. L. Kingsbury and P. L. Marston, "Mie Scattering Near theCritical Angle of Bubbles in Water," J. Opt. Soc. Am. 71, 358(1981).

55. P. L. Marston suggested that the detector be placed at or nearthe critical angle for scattering by an air bubble in water (82.9°).In P. L. Marston and D. L. Kingsbury, "Scattering by a Bubblein Water Near the Critical Angle: Interference Effects," J. Opt.Soc. Am. 71,192-196 (1981), it is explained that it is necessary tobe at or near the critical angle so that contributions from raystransmitted through the bubble adjacent to the point of specularreflection do not affect the nearly R2 dependence.

56. R. G. Holt, "Experimental Observation of the Nonlinear Re-sponse of Single Bubbles to an Applied Acoustic Field," Ph.D.Dissertation, U. Mississippi, Oxford (1988).

57. M. Strasberg, "Onset of Ultrasonic Cavitation in Tap Water," J.Acoust. Soc. Am. 31, 163 (1959).

58. R. G. Holt, "Nonlinear Dynamics of Single Cavitation Bubbles,"in Proceedings, Thirteenth International Congress on Acous-tics, Vol. 1, p. 131 (Dragan Srnic Press, Sabac, Yugoslavia, 1989);Belgrade, Yugoslavia.

59. H. W. Wyld, Mathematical Methods for Physics (Addison-Wesley, Reading, MA, 1976).

60. S. D. Horsburgh, NCPA, University, M5 38677; proprietaryunpublished software package.

61. J. H. Moore, C. C. Davis, and M. A. Coplan, Building ScientificApparatus (Addison-Wesley, Reading, MA, 1983).

62. L. A. Crum, "Measurements of the Growth of Air Bubbles byRectified Diffusion," J. Acoust. Soc. Am. 68, 203 (1980).

63. A. I. Eller, "Damping Constants of Pulsating Bubbles," J.Acoust. Soc. Am. 47, 1469 (1970).

64. L. A. Crum and A. Prosperetti, "Erratum and Comments on'Nonlinear Oscillations of Gas Bubbles in Liquids: an Interpre-tation of Some Experimental Results,' J. Acoust. Soc. Am. 75,1910 (1984).

65. The essential experimental technique used here, i.e., correlatingfluctuations of scattered light intensity from a levitated fluidspheroid to monitor the mechanical oscillations of that object,appears to have been anticipated by an earlier experiment per-formed by Marston [P. L. Marston, "Rainbow Phenomena andthe Detection of Nonsphericity in Drops," Appl. Opt. 19, 680-685 (1980)] in the investigation of forced shape oscillations ofliquid drops. The author was unaware of these results at thetime the present experiments were performed.

66. This is an acceptable criterion primarily because of the largeangle subtended by the detector. If fine structure details hadbeen important, it would have been necessary to fulfill a far morestringent criterion analogous to the far field scattering conditiondiscussed in Refs. 49, 54, and 55.


mie scattering used to determine spherical bubble oscillations

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