Mie scattering as a technique for the sizing of air bubbles

Mie scattering as a technique for the sizing of air bubbles

Gary M. Hansen

A technique has been developed which allows the radius of a bubble in a fluid to be accurately measuredwithout disturbing the bubble. Results are presented which show that radius measurements of single airbubbles in water can be achieved by shining a He-Ne laser on the bubble and measuring the light scatteredat 800. Light intensity measurements at a scattering angle of 550 were used to calibrate a photodiode detec-tion system and to demonstrate the correlation of data with Mie scattering theory. The photodiode wasthen moved to 800 for actual radius measurements. This technique was accurate to within 3% for bubbleswith radii of <80 gm.

1. Introduction

Sizing techniques which utilize Mie scattering are wellknown for the case were the particles being measuredhave an index of refraction greater than that of theirsurroundings. In fact, light scattering aerosol countershave been in use for quite some time.' Optical sizingtechniques can be put into two classes: those which aremainly concerned with determining the size distributionof a collection of particles and those which count andsize individual particles. The techniques used in bothclasses vary widely in the number of measurementsrequired. These range from intensity measurementsat a single angle2 to measurements of the entire angularpattern. 3 Those techniques which use single- or dou-ble-angle measurements are useful for in situ mea-surements but may also require additional informa-tion.2 4 5 Those that measure all or part of the angularpattern provide the most information but also requiremore extensive analysis. 3 6- 8

This study describes a single-angle technique for thecase where the index of refraction of the particle is lessthan that of its surroundings, specifically air bubblesin water. A He-Ne laser was shined on an air bubblesuspended in a standing wave acoustic field. The lightscattered at an angle of 80.00 from forward was mea-sured with a photodiode. The relative intensity wasused to achieve the radius from an equation obtainedfrom Mie scattering theory. The system was initially

The author is with University of Mississippi, Physics Department,University, Mississippi 38677.

Received 21 May 1985.0003-6935/85/193214-07$02.00/0.© 1985 Optical Society of America.

calibrated by using the light scattered at 55.00 and anestimate of the radius from a rise-velocity method.

This technique was suggested by the recent experi-ments conducted by Marston.7 8 In his experiments aphotograph is taken of the light scattered from an airbubble in water. This photo is scanned with a micro-densitometer to obtain the intensity variation withangle. Comparison of the photo pattern with the pat-tern predicted by Mie scattering theory is used to obtainthe radius of the bubble.

The need for accurate measurements of the radius ofa bubble is essential for the experimental verificationof theories of nonlinear bubble oscillations.910 Previ-ously conducted experiments11 1 2 have measured thebubble radius using a rise-velocity method which canbe in error by as much as 10%.

The rise-velocity method is the calculation of a bub-ble's radius from a measurement of its terminal rise-velocity. There are, however, certain difficulties withthis method which limit its accuracy. In general, de-termining the terminal velocity can be difficult. Forexample, in the method used by Crum,"1 the bubble isstarted from rest, and its time to rise a certain distanceis recorded. There is a reaction time involved in stop-ping the timer which can be a source of error as great as5%. There are also specific problems when using thismethod on air bubbles. First, to calculate the radiusthe drag force acting on the bubble must be known, andthere are dozens of different empirical drag laws13 thatare based on rigid spheres in which compressibility andwall effects are ignored. As a result large radius dif-ferences can be found, for the same terminal velocity,when using different drag laws. Also, it is not certainthat bubbles really follow rigid sphere drag laws. Sec-ond, bubbles that are large enough to rise continually,dissolve. For example, a 20.0-Azm bubble is observedto dissolve in -10 sec in water that is not supersaturatedwith gas. Thus the radius changes during measure-

3214 APPLIED OPTICS / Vol. 24, No. 19 / 1 October 1985

ment. Accounting for this change is difficult, since therate at which bubbles dissolve is dependent on manyparameters.14 There is also an additional problemwhen the behavior of the bubble in a sound field is beinginvestigated. To let the bubble rise freely with onlybuoyancy, drag, and weight acting on it, the sound fieldmust be turned off for the period of rise. Thus the ef-fect of this zero acoustic-pressure time must be con-sidered.

The Mie scattering technique overcomes all thesedifficulties. It is accurate to within 3%, reduciblethrough use of a more accurate method for determiningthe scattering angle; it does not disturb the bubble, iflight pressure and heating effects are ignored; andmeasurements can be made in a relatively short timeinterval limited by the speed of the electronics used.

11. Theory

A. Mie Scattering

The desired quantity in this experimental arrange-ment is the intensity of the scattered light. For an in-cident beam of unit intensity and polarized with theelectric vector parallel to the 0 = 0 plane, the intensityis given by

1 = 42 2S 2 12 COS20, (1)4ir2r2

where X is the wavelength, r is the distance from thecenter of the bubble, and 0 is the azimuthal angle. (Thenotation of Kerker 15 has been adopted.) The scatteringfunction S2 is obtained from

S2 =E + [anTn(COSO) + bn7nr(cosO)], (2)n=1 n(n + 1)

where 0 is the scattering angle, zero being forward. Theangular functions are

ir(cosP) = (coso) (

rn(COSO) = dPn (o (4)dO

where P(Ncos0) are the associated Legendre functions.The coefficients an and bn are found from

Un nWOO) - 4n(/3)04n(a)- mn(a)44/,(0)-M n(f3)44(a)

bn =M~n (e)() - /n ()n(a), (6)rn(a)Vn() - n(O)n(a)

where m = k1/k 2 = mi/M 2 is the relative index of re-fraction. Here subscript 1 refers to the medium withinthe bubble, subscript 2 refers to the medium sur-rounding the bubble, k is the wave number, M1 , m2 arethe indices of refraction, a = 2a = 27rm2 a/Xo, a =bubble radius, X0 = wavelength of the incident light invacuum, 3 = am, and primes denote differentiationwith respect to arguments. The functions

ln (a) = ajn (a), (7)

tn (e) = ath (2) (ae) (8)

are the Ricatti-Bessel functions with jn (a) and h (2) (a)

being the ordinary spherical Bessel functions of the firstand third kind, respectively.16

For a given wavelength, in a given scattering plane,and at a constant distance from the bubble, the intensityis proportional to the function I2 = I S2 2- This com-ponent is polarized parallel to the scattering plane (thescattering plane contains the incident direction and thedirection of the scattered wave) and was chosen becauseit exhibits higher scattering intensities, overall, and thefine structure is less than for the perpendicularlypolarized component.'7

For calibration purposes it was necessary to have aninitial estimate of the bubble size. For this purpose arise-velocity method was used in conjunction with thelight scattering measurements.

The rise-velocity method is the calculation of a bub-ble's radius from a measurement of its terminal rise-velocity. From the equation of balanced forces for abubble rising in water at terminal velocity U the radiuscan be calculated using18

3 U2

a =--CD,8 g (9)

where CD is the drag coefficient which is expressed asa function of the Reynolds number Re = 2a U/v, wherev is the kinematic coefficient of viscosity of the fluid andis defined as ,up, where u is the dynamic viscosity andp is the density. The empirical drag law determined bySchiller and Nauman' 3 was used for the rise-velocitycalculations in this study and was chosen because itmatched the rigid sphere standard drag curve fairly wellfor the region of interest. A rigid sphere drag law waschosen because only a 10% estimate of the radius wasnecessary for calibration purposes. The form of this lawis

24CD = - ( + 0.15 Re0 .68 7).

Re(10)

Using this drag law an iterative method was used toobtain the radius from a measure of the terminal rise-velocity.

B. Calculations

The computational scheme employed was fairlystraightforward due to the assumption that the ab-sorption was negligible. The method adopted to avoidthe difficulties with Bessel functions of large orders andarguments was a combination of that proposed byCorbato and Uretsky1 9 and Miller.20

The Mie scattering calculations were done on an IBM4341 computer in double precision. The parametersused in these calculations are as follows: index of re-fraction of the air = 1.0; index of refraction of waterrelative to air = 4/3; relative index m = 0.75; wavelengthA0 = 632.8 nm. The accuracy was checked against re-sults found in Refs. 7, 8, 15, 17, and 21.

C. Computational Results

The purpose of this study was to determine a single-scattering angle that would provide, as close as possible,a one-to-one relationship between scattered intensity

1 October 1985 / Vol. 24, No. 19 / APPLIED OPTICS 3215

6

Lnzw

I-

z

w

It

020 30 40 50 60 70 80 90 100

BUBBLE RADIUS (Am)

Fig. 1. Relative intensity of He-Ne laser light scattered from an airbubble in water at a scattering angle of 800 as a function of the bub-ble's radius. (The relative intensity of 1.0 at 50 m corresponds to

a gain = 0.76.)

and radius. A second angle was used for calibrationpurposes. The angle chosen to meet the first criterionwas 800 with 550 being used for calibration. The twoangles decided on have no other special significance.Shown in Fig. 1 is a graph of the relative intensity as afunction of radius from 20 to 100 gm for a scatteringangle of 800. Here the relative intensity of 1.0 for a50-rim radius bubble corresponds to a gain of 0.76,where gain refers to the intensity relative to that forgeometric reflection from a perfectly reflecting sphereof the same size. The overall shape of this curve wasdesirable in that the intensity rises fairly regularly (ig-noring fine structure) with radius. Figure 2 shows asimilar graph for a scattering angle of 550 (gain = 2.22).The separation of the peaks and the relative smoothnessof the fine structure were desirable for use as a cali-bration curve.

As in any experimental arrangement the intensitydetected by the photodiode is distributed over a finiteangle. For this experiment it was found that a largeaperture (1.6 mm) which would provide an acceptanceangle of 2.00 was actually helpful. A numerical inte-gration was conducted for scattering angles from 79.0to 81.00. The trapezoidal method was employed witha 0.10 step size. Results of these calculations are shownin Fig. 3. The use of a large aperture resulted in thealmost total elimination of fine structure, which is de-sirable for increased accuracy. Similar calculationswere conducted for 54.0-56.0° with a step size of 0.20.The results of these calculations are shown in Fig. 4;note again the reduced fine structure.

The functional dependence needed to use the lightintensity at 800 as a direct measure of the radius wasdetermined from a least-squares fit on the intensity datafor radii from 20 to 100 Am for the integrated results at80°. The functional dependence for this angle was

radius = 50.1(relative intensity) 0 -40 4 , (11)

with a goodness of fit of 99.994%.It is interesting to note that the general form of the

calculated intensities of Figs. 1 and 2 are, to a large de-gree, predictable from the physical optics approxima-

Wz

w

20 30 40 50 60 70 80 90 1 00BUBBELE RADIUS (m)

Fig. 2. Relative intensity of He-Ne laser light scattered from an airbubble in water at a scattering angle of 55° as a function of the bub-ble's radius. (The relative intensity of 1.0 at 50 Am corresponds to

a gain = 2.22.)

I-

U,

zjWWH

0 - ,20 30 40 50 60 70 80

BUBBLE RADIUS (m)90 100

Fig. 3. Results of a numerical integration of relative intensity ofHe-Ne laser light scattered at angles from 79 to 810 off of an air

bubble in water as a function of the bubble's radius.

4

U,zw

H

-jw

020 30 40 50 60 70 80 90 100

BUBBLE RADIUS (m)

Fig. 4. Results of a numerical integration of relative intensity ofHe-Ne laser light scattered at angles from 54 to 560 off of an air

bubble in water as a function of the bubble's radius.

tions of Refs. 8 and 17 (and references therein). Thecoarse structure at 550 (Fig. 2) arises mainly from theinterference of the reflected and refracted rays with anoptical path length difference that is small comparedto the radius of the bubble. Thus the resulting inter-


6

Fig. 5. Block diagram of experimental setup.

ference pattern changes slowly with radius, i.e., havinga quasi-period that is large compared to the wavelengthof light in water. The fine structure arises primarilyfrom the interference of the refracted internally re-flected ray and the coarse structure pattern. These rayshave optical path length differences which are ap-proximately equal to the diameter of the bubble andresult in intensities which are very sensitive to changesin bubble size. However, at 80° (Fig. 1), which is nearthe critical scattering angle, the reflected ray's ampli-tude increases greatly and dominates the coarse struc-ture as edge diffraction. The fine structure exists asbefore.

Ill. Experimental

A. Apparatus

A block diagram of the experimental setup is shownin Fig. 5. The main components were a 7-mW He-Nelaser, a cylindrical acoustic levitation cell, and a pho-todiode. In addition there were some nonspecializeditems, such as the photodiode bias supply and amplifier,and the cell's driving oscillator and amplifier. Datawere taken and analyzed with a Cyborg Corp. ISAACmodel 91A data acquisition system/Apple II computercombination.

The He-Ne laser was a model LT-7P manufacturedby Aerotech, Inc., providing 7-mW minimum outputpower at 632.8 nm in the TEMoo mode. There was a500:1 linear polarization factor and a 1/e2 beamwidthof 1.0 mm. The laser was mounted so that the E-fieldwas parallel to the scattering plane (the scattering planewas chosen to be parallel to the floor, k = 0), and thebeam traversed the vertical center of the photodiodesuspension plate.

The cylindrical cell, 64.0-mm i.d., consisted of twoceramic transducers separated by a vertical length ofglass. The cell was closed at the bottom and filled withfiltered distilled water. The transducers were driven

in one of their resonant modes at a frequency around22.0 kHz.

The photodiode was an RCA device C30957E with aresponsivity of 0.4 A/W at 632.8 nm. When radiant fluxis absorbed in the reverse-biased photodiode, the elec-tric field sweeps the created electron-hole pair out ofthe depletion region. The resulting photocurrent wasused to develop a voltage across an external 1-MO re-sistor. This voltage, which was proportional to the in-tensity of the incident flux, was then amplified by a highinput impedance dc amplifier, the output of which wasfed into the data acquisition system.

The photodiode was mounted in a custom-builtholder with a 1.6-mm aperture for an acceptance angleof 2.0° when mounted. The holder was suspended froma horizontally mounted circular plate. The center ofthis plate was attached to a voltage divider circuit. Thevoltage divider consisted of a precision linear potenti-ometer, which was fed by a constant current source.The pick-off voltage from this potentiometer was cali-brated to provide angular information.

B. Methods

Since the scattered light intensity was critically de-pendent on the scattering angle, an accurate determi-nation of the photodiode angle was essential. This taskwas undertaken as the first step in the experiment. Thezero scattering angle was defined as forward and wasdetermined from a weighted average over a beam profileobtained by stepping the photodiode with a motor fromone side of the beam to the other, voltage readings beingtaken at each step. Two independent methods werethen used to determine the correlation between po-tentiometer voltage and scattering angle. The firstmethod consisted of mounting a spectrometer at thevertical center of the photodiode suspension plate. Thetelescope of the spectrometer was then used to sight theangles. The potentiometer voltage for each angle wasread from a digital voltmeter, and the correlation wasmade by a least-squares fit from several angles. Thesecond method consisted of mounting a diffractiongrating on the spectrometer, shining the laser throughthe grating, and scanning the photodiode through thepattern. Matching the measured locations of the peaksto the angles given by theory gave the relationship be-tween angle and voltage. The two methods agreed towithin 0.8% and provided angular information accurateto within 0.25°.

The levitation cell was then set in place and alignedto be in the vertical center of the plate. A standing waveacoustic field was established in the cell so that an an-tinode existed in the center. This was achieved byvarying the water level in the cell as well as the fre-quency. The radiation force on the bubble in the centerof the cell was then used to balance the buoyancy forceand effectively levitate the bubble, causing it to remainrelatively stationary. Bubbles were initially inducedin the cell by increasing the acoustic-pressure amplitudeuntil gaseous cavitation occurred and then reducing theamplitude to allow for coalescence into a single bubble.This levitation technique also allowed for the adjust-


ment of the bubble position up and down through aslight variation of the pressure amplitude.

The voltage that corresponds to the light intensitywas obtained by manually triggering the ISAAC/Appleto take 100 voltage readings while the bubble was beingacoustically held in the center of the laser beam. Thebackground voltage was then subtracted from the av-erage of these readings to yield one data point.

The rise-velocity radius was obtained by using asighting telescope that had graduated graticule marks.The actual distance that the bubble traveled was knownin relationship to these marks. With the bubble levi-tated in the center of the field of view of the telescopethe sound field was turned off; simultaneously, a timerwas turned on. The bubble was allowed to rise a fewmarks, the timer was stopped, and the sound field wasturned back on. The time and distance of the rise gavethe rise-velocity assuming negligible time to reach ter-minal velocity. The radius was then determinedthrough an iterative calculation.

Little time was allowed to elapse between the lightintensity reading and the rise-velocity determination.This was done in an attempt to assure that the radiusdid not change significantly between the two methods,since bubbles between 20 and 80 Am change size at arate of the order of 0.3 Am/sec in low amplitude soundfields. 14

For consistency and accuracy, certain procedureswere always followed. First, the photodiode readingswere triggered only when the bubble was in a specifiedlocation, as viewed through the sighting telescope.Second, the rise time for bubbles was kept between 1and 2 sec whenever possible. Times <1 sec showedreaction time errors which were noticeable in the largebubble data scatter (see Fig. 6). Times >2 sec allowedthe bubble to dissolve too much. This effect was moreprevalent in smaller bubbles. A 20.0-gm bubble wasobserved to dissolve in -10.0 sec. Third, attempts weremade to avoid parallax errors in noting rise distances.

The water was distilled and filtered through a 7-gmfilter to eliminate as many background scatterers aspossible. The water used was also slightly degassed toreduce growth rates. The glass opposite the photodiodeand at 0 was covered (inside the cylinder) with blacktape to reduce reflections. Experiments were con-ducted with the room lights off, and each run was lim-ited to 100 data points to reduce the fatigue error, whichincreased reaction time.

C. Procedure

The photodiode was positioned at 550, and rise-ve-locity and light scattering intensity data were taken.The raw values for light intensity were divided by thevalue which was achieved for a rise-velocity radius of 50gim. The cumulative results are given in Fig. 6. Thesedata match that of the theoretical predictions of Fig. 4except for a shift in radius. The spread of the datapoints gives the best indication of the experimentalerror. When the theoretical results and data werecompared it was found that the rise-velocity methodgave values for the radius that were 10% too small at -50

4 I1

I-

0

20 30 40 50 60 70 80 90 100RADIUS BY RISE-VELOCITY METHOD (m)

Fig. 6. Experimental values of the relative intensity of He-Ne laserlight scattered from an air bubble in water at an angle of 550 plottedagainst the radius of the bubble obtained by the rise-velocity

method.

4

U,zw

0 20 30 40 50 60 70 80 90 100

BUBBLE RADIUS (m)

Fig. 7. Experimental values of the relative intensity of He-Ne laserlight scattered from an air bubble in water at an angle of 55° with the

data shifted to match the theoretical curve at 56.9 um.

gim. Figure 7 shows this discrepancy. In this figure thevalue of the rise-velocity radius at 51.7 gm was forcedto match the value of the theoretical curve at 56.9 gim.This same 10% shift in radii was carried through for therest of the experimental rise-velocity values. A cor-rection was also made for the shift in relative intensitydue to the initial error in radius. The match of the datawith the theory indicated that this was indeed the cor-rect shift. Since the form of the drag law is not linearthe actual shift of all radii may not be 10%, but theoverall fit indicated that this was a good first approxi-mation.

Similar measurements were made with the photo-diode shifted to 80°. The same 10% shift in the rise-velocity method radii was used; the data are plottedalong with the theory in Fig. 8. The close agreementbetween the experimental points and the theoreticalcurve indicates that the 10% shift was real and verifiesthat the light intensity reading at the rise-velocity ra-dius of 45.5 m is the value to be used as the denomi-nator for calculating the relative intensity. The lightintensity readings at 80° were then divided by this de-nominator and used to calculate the radius directly.


>

TI

U,

I-

W .

020 30 40 50 60 70 80 90 100

BUBBLE RADIUS (m)

Fig. 8. Experimental values of the relative intensity of He-Ne laserlight scattered from an air bubble in water at an angle of 80° withradius values obtained by a rise-velocity method shifted by 10%

plotted along with the theoretical curve.

IV. Discussion

The results of this study showed that accurate radialmeasurements of single air bubbles could be achievedwith a Mie scattering technique. Using the relativelight intensity scattered at 80°, Eq. (11) could be usedto obtain the radius. This was indicated by the closeagreement between the, theory and the corrected rise-velocity data in Fig. 8. This method gave the radius ofthe bubble to within an estimated 3% for bubbles ofradii of <80 Am.

The estimated 3% was a summation of the possiblesources of error in the light scattering measurements.The major source (as much as 2.0%) was the 0.25° anglemeasurement error. The next largest source was in thevoltage readings. For 100 readings the standard de-viation divided by the mean was -1.0%, which trans-lates into a <1% radius error for the entire range ofbubble sizes. These fluctuations were probably due tothe oscillation of the bubble, background scatterers,changing bubble size, and noise in the electronics. Thenumerical integration and the acceptance angle mea-surement added the next most significant portion.

The other small sources of error fell into two catego-ries: those due to assumptions made prior to calcula-tions and those due to the experimental arrangement.Four assumptions were made prior to the computations.First, the index of refraction of the water relative to theair was assumed to not vary significantly from 4/3.Second, absorption of the light by the air in the bubblewas ignored. Third, the bubble interface was assumedto be pure air-water; i.e., surface contaminants wereignored. Fourth, the focusing caused by the cylindricalgeometry did not affect the theoretical assumption ofplane waves, since the distance from the bubble to thefocus point was greater than a thousand bubbleradii.2 2

Some small errors were inherent in the experimentalsetup. A small component of the perpendicular fieldmay have been detected. This component could arisefrom inaccurate horizontal alignment of the E-field.Inaccurate leveling of the laser beam could change the

scattering plane which would introduce a 0 component.This problem could also arise from a deviation in theglass cylinder. A source of error that was probably mostprevalent in large bubbles (>80 gim), which requirehigher acoustic pressures to hold them in the beam, isthe fact that bubbles oscillating with a great enoughamplitude have an average radius that is larger than thestationary radius.9 The radius measured was assumedto be the average radius, which was equal to the sta-tionary radius, since the bubbles were oscillating at 22kHz, and the 100 voltage readings were taken at a muchslower rate. This effect may have been a minor causeof the data scatter in the large bubble region.

Another source of error was any nonlinearity in thephotodiode's response curve. A rough check of thephotodiode linearity was conducted. The results didnot indicate that the response was nonlinear over therange of input intensities used in the actual measure-ments. This nonlinearity could account for some of thedeviation of the experimental points from the theory.Another source of the deviation is the short time dif-ference between light intensity and rise-velocity mea-surements. Also any change in the temperature of thewater during a data run was ignored.

Although this Mie scattering method is theoreticallyfeasible for all bubble sizes, there was an exceedinglylarge amount of data scatter for large bubbles. Thislack of correlation was assumed to be a result of exper-imental errors in the rise-velocity measurement sincethe Mie scattering theory has been shown to be accu-rate.7 8 The experimental error was greatest for largebubbles because the reaction time errors were larger forlarge bubbles and the large bubbles were more difficultto keep stationary in the laser beam.

Despite the lack of correlation for large bubbles theMie scattering method was superior to the rise-velocitymethod for measuring the radii of air bubbles. The Miescattering method meets all the requirements of atechnique to be used for bubble dynamics measure-ments. First, it was precise (estimated 3% error).Second, it did not disturb the bubble, if optical radiationpressure and heating effects are ignored. Third, it wasfast; once calibrated it required measurement of onlya single parameter.

This work was supported by the Office of Naval Re-search and the National Science Foundation. Theauthor also wishes to thank L. A. Crum and P. L. Mar-ston for their helpful comments.

References1. J. R. Hodkinson and J. R. Greenfield, "Response Calculations

for Light-Scattering Aerosol Counters and Photometers," Appl.Opt. 4, 1463 (1965).

2. D. Holve and S. A. Self, "Optical Particle Sizing for in situ Mea-surements. Part 1," Appl. Opt. 18, 1632 (1979).

3. H. H. Blau, Jr., D. J. McCleese, and D. Watson, "Scattering byIndividual Transparent Spheres," Appl. Opt. 9, 2522 (1970).

4. W. M. Farmer, "Measurement of Particle Size, Number Density,and Velocity Using a Laser Interferometer," Appl. Opt. 11, 2603(1972).


5. S. Boron and B. Waldie, "Particle Sizing by Forward LobeScattered Intensity-Ratio Technique: Errors Introduced byApplying Diffraction Theory in the Mie Regime," Appl. Opt. 17,1644 (1978).

6. T. W. Alger, "Polydisperse-Particle-Size-Distribution FunctionDetermined from Intensity Profile of Angularly Scattered Light,"Appl. Opt. 18, 3494 (1979).

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

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

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

10. M. Fanelli, A. Prosperetti, and M. Reali, "Radial Oscillations ofGas-Vapor Bubbles in Liquids. Part I: Mathematical Formu-lation," Acustica 47, 253 (1981).

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

12. L. A. Crum and A. Prosperetti, "Nonlinear Oscillation of GasBubbles in Liquids: An Interpretation of Some ExperimentalResults," J. Acoust. Soc. Am. 73, 121 (1983).

13. R. Clift, J. R. Grace, and M. E. Weber, Bubbles, Drops and Par-ticles (Academic, New York, 1978).

14. R. K. Gould, "Rectified Diffusion in the Presence of, and Absenceof, Acoustic Streaming," J. Acoust. Soc. Am. 56, 1740 (1974).

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

16. M. Abramowitz and I. A. Stegun, Handbook of MathematicalFunctions (Cambridge U.P., London, 1922).

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

18. L. M. Milne-Thomson, Theoretical Hydrodynamics (Macmillan,New York, 1968).

19. F. J. Corbato and J. L. Uretsky, "Generation of Spherical BesselFunctions in Digital Computers," J. Assoc. Comput. Mach. 6,366(1959).

20. J. C. P. Miller, British Association for the Advancement ofScience, Mathematical Tables, Vol. 10, Bessel Functions, Part2 (Cambridge U.P., London, 1952).

21. H. C. van de Hulst, Light Scattering by Small Particles (Wiley,New York, 1957).

22. H. Chew, M. Kerker, and D. D. Cooke, "Light Scattering inConverging Beams," Opt. Lett. 1, 138 (1977).

Of OpticsticsThe Managing Editor welcomes news from any source. It should be addressed toP. R. WAKELING, WINC, 1613 Nineteenth Street N. W., Washington D. C. 20009

SCIENCE IN SMALL SCHOOLS

Reports on science studies in small schools appeared in two June issues of The New York Times. They arereprinted here by permission. The New York Times Company.

Forty-eight colleges seek science supportGene 1. Maeroff

Forty-eight colleges renowned for turning out future law-yers, physicians, corporate executives and government offi-cials, have banded together to publicize and seek support fora role for which they are less known: the production of sci-entists. Led by Carleton, Franklin and Marshall, MountHolyoke, Oberlin, Reed, Swarthmore and Williams Colleges,the presidents of most of the institutions met in Oberlin, Ohio,last Monday to discuss the implications of a report that saidthey had considerable success in training scientists. "Someof us knew we were doing a good job in science, but we wereunaware of just how important these institutions were as agroup," said John S. Morris, the president of Union Collegein Schenectady, N.Y., one of the forty-eight.

The presidents are concerned that the expense of teachingand research in the basic sciences is rising so fast that theposition of their colleges is now imperiled. They are partic-ularly chagrined over figures showing that their group of in-stitutions gets less than one-half of 1% of the funds alloted bythe National Science Foundation. The report, "EducatingAmerica's Scientists: The Role of the Research Colleges," was

prepared by researchers at Oberlin who identified liberal artscolleges whose science programs attracted many well-qualifiedstudents. Among the findings were these:

Twenty-eight percent of the freshmen at the forty-eightschools plan to major in basic sciences, such as biology,chemistry, physics and mathematics, as against 14% at allhighly selective private colleges and 6.6% in all of higher ed-ucation.

From 1975 to 1984, when the portion of students intendingto major in science declined by 33% in all of higher education,it fell by only 12% in the forty-eight schools.

The portion of students graduating from the forty-eightcolleges with degrees in science remained steady at 24% from1975 to 1983, while the portion throughout the country fell to7.7% from 9.4%.

The number of science baccalaureates awarded by thesecolleges from 1976 to 1980 rose 16%, while the number of un-dergraduate science degrees granted by the twenty highest-rated universities dropped 14%.

"Leading liberal arts colleges rank at or near the top of allAmerican institutions of higher education-including mul-tiversities and major centers of research-in the training of

continued on page 3226


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Mie scattering as a technique for the sizing of air bubbles