Aerosol measurement by laser doppler spectroscopy—I. Theory and experimental results for aerosols homogeneous

Aerosol Science, 1972, Vol. 3, pp. 501 to 514. Pergamon Press. Printed in Great Britain.

AEROSOL M E A S U R E M E N T BY LASER D O P P L E R SPECTROSCOPY--I. T H E O R Y A N D E X P E R I M E N T A L RESULTS F O R AEROSOLS H O M O G E N E O U S

W . HINDS a n d P. C. REIST

Harvard School of Public Health 655, Huntington Avenue, Boston, Massachusetts 02115, U.S.A.

(Received 13 April 1972) Abstract--The basic theory, experimental techniques and results are presented describing a technique for sizing aerosol particles in situ using laser Doppler spectroscopy. Unlike conventional light scattering procedures which use average intensity information, this technique utilizes the Doppler shifted frequency of the scattered light produced by the Brownian motion of the aerosol particles to determine particle diffusion coefficients and size. Experiments were carried out using monodisperse dibutyipthalate aerosols and monodisperse polystyrene latex spheres, in concentrations ranging from 10z to l0 s particles per cubic centimeter. Measured particle sizes were within 10 per cent of the size predicted by conventional light scattering methods for the DBP particles and the reported sizes of the PSL particles. Based on these results it is concluded that laser Doppler spectroscopy can be Utilized to accurately measure aerosol particle size in situ.

INTRODUCTION

LIGHT scattering techniques find application in a variety of aerosol sizing situations. All use average light intensity measurements, usually as a function of wavelength, scattering angle or polarization angle, and all are subject to the same inherent limitations. This paper describes a fundamentally different optical technique for sizing aerosols - - that of analyzing the frequency information present in the scattered light. The technique, sometimes referred to as light beating or photoelectric mixing spectroscopy, can best be described as laser Doppler spectroscopy or LDS. When aerosol particles are illuminated with monochromatic light, their Brownian motion gives rise to frequency modulations of the scattered light. By measuring and analyzing these frequency modulations one can determine the distribution of Brownian velocities and hence the diffusion coefficient of the aerosol particles. Using the StokesEinstein relationship with the appropriate slip correction particle size can thus be determined. Laser Doppler spectroscopy became practical with the development of the continuous wave laser in the 1960's. It is the extremely monochromatic light emitted by a laser that makes LDS sensitive enough to detect the minute frequency shifts resulting from the random motion of particles within the light beam. As the particle moves within the illuminated re#on, the frequency of its scattered light will be shifted a very small fractional amount due to the Doppler effect. As might be expected, the random motion the particles undergo gives rise to a whole spectrum of frequency shifts. A plot of the scattered light intensity versus 501

502

W. HINDS and P. C. REts'r

frequency gives a curve having the characteristic shape of a Lorentzian curve. The width of this Lorentzian curve can range from a few hertz to a few kilohertz compared to the fr~luency of the incident light which is on the order of I0 ~4 Hz. Thus to measure the shape of the spectrum of frequency shifts due to Brownian motion a technique with an extraordinary fractional resolution, approaching one part in 10 t4, is required. High resolution mcasurerncnt is accomplished with LDS by translating the spectrum originally at 10~4 Hz down to the low audiofrequency range where its shape can easily and accurately be determined by a conventional audio frequency spectrum anatyser. This translation from optical to audio frequencies can be accomplished by either of two light beating techniques, heterodyne or homodyne detection. Both techniques are direct analogs of the mixing of ac electrical signals in superheterodyne and monodyne receivers. In heterodyne detection the scattered signal is mixed o r " beat" with a strong '" local oscillator" in a photomultiptier tube. If the original unshifted laser light is used as the local oscillator then this process will exactly translate the spectrum from optical frequencies to zero frequency. Similar results can be obtained using homodyne detection. In this technique each element in the optical spectrum (if the optical spectrum can be thought of as divided into a series of discrete frequencies) is allowed to beat with every other element in the spectrum. The net result of this ~lf-l~ati~ process is to exactly reproduce the optical spectrum at zero frequency exc.cpt that it will be twice as wide. A schematic version of both detection schemes is shown in Fig. 1. Unlike homodyn© spectroscopy which measures the motion of particles relative to each other, heterodyne spectroscopy measures absolute motion, hence it can be used to measure local velocity in a moving fluid. Ymt and Ctaa~Ns (1964) showed that light scattered by impurities in a fluid is Doppler shifted by an amount proportional to local fluid velocity. Traditional light scattering measurements assume that the frequency of the scattered light is the same as that of the incident light; so-called elastic scattering. Because small i}

Laser spectrum - - . Particle

>,

? Heterodyne

Pho'tocurren f

~1014 HZ OpticoI

Photocurrent

~ 0 ) 4 HZ Opticot

0

~

Homodyne

FIG. 1. S e P t i c

representation of heterodyne and homodyne spectroscopy.

Aerosol measurement by laser Doppler spectroscopy--I

503

shifts in the frequency of the scattered light do take place and form the basis of light beating spectroscopy, this type of scattering is called quasi-elastic to distinguish it from inelastic scattering processes such as Brillion and Raman scattering. PECORA (1964) published a detailed theoretical analysis of the light scattered from an assemblage of particles undergoing Brownian motion and predicted that the scattered light would have a Lorentzian-shaped frequency spectrum. CUMMINSet al. (1964) observed the spectrum of scattered light from a water suspension of polystyrene latex spheres using heterodyne spectroscopy. At about the same time FORD and BEt,reDEK (1965)first used homodyne detection to study critical point phenomena. Pecora's predictions for homodyne spectroscopy were found to be correct by ARECCHI et al. (1967) using polystyrene spheres in water and DUBIN et al. (1967) using biological macromolecules. Comprehensive review articles of this procedure have been written by BENEDEK (1969), CHU (1970), CUMMINS and SWINNEY (1969) and FRENCH et al. (1969). However, no experimental data have been available showing the applicability of this technique to the study of aerosols. THEORY

The following is a summary of the theory describing the expected frequency spectrum of an aerosol LDS experiment. We consider only the translational diffusion of a monodisperse spherical aerosol. As shown in Fig. 2, the aerosol is illuminated by a plane monochromatic wave and scatters light, (assumed to be spatially coherent over the entire detector surface) through an angle 0 to the photomultiplier tube. The field scattered by the jth particle is given by VAN DE HULST (1957) as, Ej(t) = Es exp[i(K.r~(t) - - o~0t)],

(1)

where Es is the amplitude of the scattered field, to0 is the frequency of the incident light, rj(t) is the position vector of the jth particle, and K is the scattering vector given by, K = Ko -- Ks.

(2)

As shown in Fig. 2 the magnitude of the scattering vector for/Co ~ Ks is K = 2Ko sin(0/2), where Ko = 2~r/A.

/ Laser

FIO. 2. Diagram showing scattering geometryand the scattering vector.

(3)

504

W. HINDSand P. C. REIST

As an aerosol particle moves in the iUuminated field the phase factor, K.rj(t), changes and produces fluctuations in the scattered field that mirror the movement of the particle. Only those particle motions with a component in the direction of K will produce any change in .4,-.). the phase factor. When K.rj(t) changes by 2rr in one second it produces a change in the frequency of the scattered field of 1 Hz. The phase factor will change by 2~- when the particles move a distance of 2~r/K in the direction of K. Thus the value of K, which is proportional to sin(0/2), d©termines the " sensitivity " of the frequency shifting process. The total scattered field reaching the detector will be the sum of the fields from each of the N particles in the illuminated region; N

"~ '4)

E,(t) = ~ E, exp[i(K.rj(t) + ~%t)].

(4)

J--1

Rather than attempt to express the exact temporal variations of the phase factor for Brownian motion we use the correlation function, which for the scattered field has the form:

Re(z) = E*(t) E(t÷~-) ,

(5)

where the overline denotes an ensemble average and the * indicates the complex conjugate. This function is the ensemble average of the product of two quantities measured at slightly different times, t and t q- r. For a stationary Mark.off process, which we consider here, R0") is not a function of t, but only of t h e time difference, ~-. The Wiener---Khinchine theorem, (Krr~L, 1958), provides the relationship between the correlation function of a random process and its frequency spectrum;

S(~o) ---- ~ J

ei'°~R(r) d r ,

(6)

--=(5

where S(oJ) is the spectral density, Thus, if we can determine the correlation function as defined by equation (5) we can directly calculate the corresponding spectrum by equation (6). From equations (4) and (5) we get the correlation function for fluctuations in the scattered field; Re(~-) = E,z e-i°'o~ ~ ~ exp [ - iK.rj(t) -- iK.rj(t ÷ r)]. k j

(7)

Here we make several simplifying assumptions. First, we assume that the motion of thejth particle is in no way correlated with the motion of the kth particle. If the particles are uncorrelated then the correlation function will be zero when different particles are considered, that is whenj ~ k. Second, since the motion of any particle is equivalent to that of any other particle, the ensemble average in equation (7) is the same for any value of j, hence, the ensemble average of the sum is simply N times the ensemble average of a single particle. The change in particle position over time interval -r, Ar(r), depends only on r so we let Ar(r) = r(t) -- r(t + r). Finally, we note that N Es z is simply the average intensity at the detector, h. We can now rewrite the correlation function, equation (7), as:

Re(q) = h e -i°J0' exp [iK'Ar(r)].

¢8)

The ensemble average exp [iK.Ar(r)], represents the integral of exp [--iK.Ar(r)] over the conditional probability distribution P(r, t[0, 0). Such a probability distribution describes

Aerosol measurement by laser Doppler spectroscopy--I

505

the likelihood of a particle being at r at time t if the particle was at r = 0 when t = 0. Thus, P(r, tl0, 0) is equivalent to P(Ar, r), and the ensemble average is given by, el,

exp [--i K.Ar(r)] = f

P(Ar, r) exp [--i K.Ar(r)] d Ar.

(9)

This probability distribution is given by Fucns (1964) as,

P(Ar, r) = (4~rDr) -1/2 exp(-- Ar2/4Dr),

(10)

where D is the particle diffusion coefficient and Ar includes many steps of a random walk. Substituting equation (10) into (9) and integrating gives exp[--i K.L~r(t)] = exp(--K2Dw),

(11)

and the correlation function becomes, Rg(r) = Is e-i~0 ~ exp(--K2V~').

(12)

We now obtain the spectrum of the scattered field using the correlation function [equation (12)] and the Wiener-Kintchine theorem [equation (6)];

Se(oo) = 1~ K2D 7r (KZD) 2 + (co --

(t3) OJo)2

Equation (13) has a Lorentzian form (see Fig. 7)centered at the laser frequency with a half width at half maximum (HW) of K2D rad/sec. To get the spectrum of the scattered light intensity we need the intensity correlation function which is given by MANDEL (1963) using the field correlation function as,

Rl('r ) = I32 + [R£(~')[2 .

(14)

Rr(~-) = 132 [1 + exp(--2K2Dr)],

(15)

Equations (12) and (14) give,

which gives, disregarding a d.c. term, an intensity spectrum of, St(c°)

I~ (2K2D) rr (2K2D) 2 + oJ2

(16)

Again the spectral shape is Lorentzian, but this Lorentzian is centered at w = 0 instead of o, ~ 1014 as was the spectrum of the scattered field, equation (13). Also note that the spectral width is twice that of the scattered field, thus the half width at half height for the intensity fluctuation spectrum is HW = 2DK 2, rad/sec or, DK 2 HW = Hz.

(17)

The diffusion coefficient of the illuminated particles is directly proportional to the width of the intensity spectrum, now at zero Hertz. Thus far, we have made no assumptions about the detection technique; we have determined only the spectral shape of the scattered field and intensity. Any detector that responds to intensity will yield a fluctuation spectrum whose

506

W. HINDSand P. C. R~tsr

width is DK2/rr Hz. The human eye can detect fluctuations in the scattered light intensity when D K 2 is sufficiently small ( ~ 1 Hz). This is an example of homodyne detection where the mixing process involves only the scattered light. The photocurrent of a photomultiptier tube, PMT, is proportional to the intensity of light incident upon it and will have a fluctuation spectrum similar to the scattered intensity spectrum. It is not exactly the same because the photoemission process produces a series of very brief (10 -9 SCC) pulses which give rise to a shot noise which is constant at all frequencies. The photocurrent is i(t) = fl [(t),

(18)

where fl is a suitably defined quantum ¢tfacieney, From equations (15), (16) and (18), we get the photoeurrent fluctuation spectrum which is given by, iz (K2D/lr) St(f) ---- ~ (K2D/~,) 2 ÷ f 2 + S.N. for f >

0,

(19)

where f is in Hz, the d.c. term has been dropped, and S.N. is the shot noise term which depends only on the average intensity. The essential fc~ttums of this spectrum are shown in Fig. 7. Using equation (3) we can express the half width as, 16D~HW = ~ sin 2 0/2 Hz;

(20)

lP

note that the HW is proportional to D and to sin 2 (0/2), and does not depend on particle refractive index. Measurement of the spectral shape and HW enable determinations of the particle diffusion coefficient by equation (20). Then, using the Stokes-Einstein relationship, kTCc D = ~ 31r r/d

(21)

particle diameter can be computed. The slip correction factor, Co, prevents equation (21) from being solved explicitly for the particle diameter. However, by combining equations (20) and (21) a plot of HW versus particle diameter for selected values of 0 can be obtained as shown in Fig. 3. Such a plot for a series of scattering angles provides immediate deterruination of the particle diameter based on the measured HW. In practice most continuous wave lasers operate multimode, that is, the light consists of a family of very narrow freqtmncy modes separated from each other by ~ l0 s Hz. This does not affect the spectrum because the intermodal beat notes are well beyond the frequency range used in LDS and each mode produces essentially identical spectra which simply add in the detection process. EXPERIMENTAL

The minimal requirements for homodyne spectroscopy are; an assemblage of particles illuminated by coherent light, a detector to convert the scattered light fluctuations to an electrical signal, and a means of measuring signal power versus frequency. The basic components for this homodyne spectroscopy experiment are shown in Fig. 4. Particles, illuminated by a laser beam, scatter light to a photomultiplier tube. The photo-

Aerosol measurement by laser Doppler spectroscopy--I

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Fla. 3. Theoretical spectrum half=width versus particle diameter for scattering angles of 10, 30, 60, 90 and 180°. • . • ' ". .". ".

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onalyser

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FIG. 4. Block diagram of homodyne spectrometer. multiplier signal, becomes the input to the spectrum analyser, an analog device whose output is a voltage proportional to the square root o f t h e instantaneous r.m.s, signal power at a given detection frequency. The spectrum analyser output is integrated over a timed interval and displayed by a digital voltmeter. The spectral shape and half width are determined from a series of such measurements over a suitable frequency range.

508

W. HINDSand P. C. REIST THE OPTICAL SYSTEM

All optical components, shown within the dashed region in Fig. 4 were contained in a 70 x 100 x 25 cm light tight plywood box, details of which are shown in Fig. 5. The box was mounted on a --~ 100 Kg marble slab, supported by partially inflated inner tubes to isolate the optical system from building vibrations. The P M T was fixed in position and collimated on the vertical aerosol stream centered above the pivot point of the alignment arm. The alignment arm had adjustable front surface mirrors at each end so that various scattering angles could be obtained by aligning the laser beam over the alignment arm axis. For some runs a focused beam was needed and a focusing lens and aperture were positioned on the alignment arm to give a focused beam at the pivot point. This apparatus provided a range of scattering angles from 0 to -,~ 170 ° with an accuracy of 0.1 °. The laser, a 5 mW H e - N e Spectra Physics model 120, was mounted on screw adjustable feet and pivoted at one end. This laser provides vertically polarized red light with a wave length of 0.6328 tzm and was used without any additional amplitude stabilization. The manufacturer gives the beam diameter as 0-65 mm with a divergence t .7 miUiradians. An RCA 7265 photomultiplier tube was used. This tube has a quantum efficiency of approximately 5 per cent for the red light of the H e - N e laser. The tube has a light-tight covering with a 0-5 mm aperture at its end window. A 108 mm focal length lens was permanently positioned with the P M T aperture at its focal point and aligned with the pivot point of the alignment arm. This allowed only parallel rays of a given scattering angle to reach the photocathode. AEROSOL GENERATION AND FLOW SYSTEM Two methods of aerosol generation were used to produce the test aerosol; vapor condensation of high boiling point liquids, and nebulization of polystyrene latex (PSL) spheres. The condensation generator is a modified Weinstein-Rappaport type with a reheater section and a provision for adding additional nuclei from a nichrome wire. This generator produced a high concentration, up to 106/cm 3, of homogeneous aerosol using dibutylpthalate DBP. The aerosol flow was controlled by a flow divider, one leg of which provided the Light tight

box

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trap

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PMT t

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_ 1. . . .

~ 8

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FIG. 5. Plan view of optical system.

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Aerosol measurement by laser Doppler spectroscopy--I

509

appropriate aerosol flow rate for the laser spectroscopy experiment. Higher order Tyndall spectra (H.O.T.S.) and the polarization ratio for 90 ° scattering were used for sizing this aerosol, SINCLAIRand LA MEP, (1949). The nebulizing of polystyrene latex spheres (Dow Chemical Company, Midland, Michigan) provided aerosols with diameters of 0-557, 0-796, and 1.10 /~m. A Collison nebulizer operated at 10 to 30 psi or a deVilbiss D40 nebulizer operated at 5 to 15 psi suspended the particles in a fine spray which was swept through a crude impactor to remove the largest droplets, then dried by mixing with dry air. Suspension concentrations of 0.05 to 0.5 per cent solids in distilled water were used which gave aerosol concentrations of about 104/cm a and 5-20 per cent doublets. This high percentage of doublets was permitted because of the necessity of obtaining a high overall particle concentration for good detectability. The concentration of doublets was determined for each run by electron microscopy and corrections were made in the analysis as explained below. The PSL suspensions were put in an ultrasonic bath prior to use to break up any agglomerates in suspension. Figure 6 shows the aerosol flow system, the aerosol flows down the center tube accompanied by sheath air at the same velocity in the outer tube. The two streams traverse the illuminated gap and are collected below by the exhaust line. Open cell foam is used in the sheath air annulus and collector tube to ensure a uniform velocity distribution. A flow range of 50 to 400 ml/min was used in these experiments. The laminar aerosol stream was perpendicular to the scattering plane a.nd as such had no flow velocity component in the direction of the scattering vector, K; consequently the aerosol stream velocity made no contribution to the resulting homodyne spectrum.

-}

~ 16rnm~

Aerosol

inlel"

~i"li;i. [ ;:i.:"):::!.:.;::i.--i:i .:!.:::.::,i.".:.~.:.:; ::.;.:Z: 1E x h o , , s *

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,

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, / / / / / / / / / I " , / /

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FIG. 6. Sectionalview of aerosol flow system.

510

W.H~Dsand P. C. R.Emr

A dynamic flow system was used rather than a static container because of the necessity of maintaining constant concentrations throughout a run of 5 to 30 minutes. Also, problems of elastic scattering from the aerosol container (which can cause heterodyning) or scattering angle corrections due to interface refraction, are eliminated by having the scattering take place in a containcrless stream. Care was taken to adjust the flow so that the aerosol stream did not diverge or converge while passing through the laser beam, since this would introduce velocity gradients in the direction of the scattering vector. In addition, the flow was adjusted such that, residence time in the beam > > 1/HW, since too high a velocity through the beam gives a signal pulse which distorts the low frequency part of the spectrum. ELECTRONIC

SYSTEM

The basic electronic components are shown in Fig. 4. The spectrum analyser is a type 3L5 Tektronix plug-in module used in conjunction with a 564 Tektronix oscilloscope. The oscilloscope was used in manual sweep mode and served only as a calibrator and power supply. The frequency to be analysed and the detector bandwidth were set manually with the spectrum analyser controls. These fixed frequency measurements were found to be more satisfactory than the slow sweep and plotting method, because all the sampling time was spent collecting information that is directly used in fitting the data to a Lorentzian curve. The spectrum anaiyser output was integrated by a signal integrater over a time interval controlled by a remote timerl The electronic system has a frequency response that was flat to 45 kHz. OPERATION

About 10 sampling frequencies ranging from about 0.5 to 5 times the expected H.W. were selected and the spectrum analyzer resolution set to less than 0.2 times the expected HW. Signals were integrated over 15 second intervals at each of the sampling frequencies and also at 45 kHz, the latter giving an estimate of the shot noise of the system. The series of samples were repeated about 5 times since multiple sampling at each point minimized the effects of instrument drift. These data were then fit by a least square method to a Lorentzian curve using an electronic computer, and the half width at half height determined. Finally, the particle diameter was found from a plot of HW versus particle size for the scattering angle used, as shown in Fig. 3. EXPERIMENTAL

RESULTS

AND

DISCUSSION

The spectrum resulting from a typical run is shown in Fig. 7 as a plot of spectral density versus frequency. This plot was for a DBP aerosol with 105 to 106 particles per cm 3 and a diameter estimated by the angular position of the H.O.T.S. of 0.34/~m and by 90 ° polarization ratio of 0.33/~m. The corrected data points are shown and the curve is the best fit Lorentzian with a half-width of 584. Hz and a r.m.s, error in fit of 0.75 per cent. This HW corresponds to a particle diameter of 0.35 t~m. for the scattering angle used of 25 °. The data can be seen to be accurately Lorentzian even out to several half-widths. Figure 8 shows the measured half-widths of two DBP aerosols plotted against the K 2 values [K2oc sin2(0/2)]. The scattering angles used, 15°, 20 °, 25 °, 30 °, and 35 °, are indicated at the appropriate K 2 value for A = 0.6328 /~m. The data closely fit straight lines through the origin with correlation coefficients greater than 0.999. The slope of the

Aerosol measurement by laser Doppler spectroscopy--I

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-*2- 0 - 2

5 8 4 Hz o

S(f)= 0 3 3 9

i.(f/584)2

+0

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Frequency,

FIG. 7. H o m o d y n e spectrum for a dibutylpthalate aerosol at a scattering Angle o f 25 °. The me,~ured dam are shown as circles and the best fitting Lorentzian for them is shown by the curved line; the r.m.s, error in fit is 0.75 per cent.

I000

800

N

-r

60o

J: 1:3

o

"r

4OC

200

131 Z5 °

20'

25 °

K2'

PIG. 8.

30 °

35 °

cm-2x i0 -9

Half-width versus K a for two dibutylpthalate aerosols. Scattering angles used are indicated at the appropriate K 2 values. The lines are regression lines through the origin.

512

W. HINDS and P. C. RE/ST

upper regression line corresponds to 0.35 /~m diameter which can be compared with an average diameter as predicted by H.O.T.S. of 0.37 ~m and by the 90 ° polarization ratio of 0.36 ~m. For the,lower line the measurements show a greater range, 0.61 ~m for H.O.T.S. and 0.55/~m for LDS. Thus, the spectral width shows the predicted K 2 dependence and predicts the aerosol size within 10 per cent of the size predicted by conventional light scattering. Although these results show good agreement with sizes predicted by conventional light scattering measurements both depend on light scattering and may be equally biased from the true size. More convincing evidence that HW oc D comes from the PSL results as shown in Fig. 9. These data have been corrected for the presence of a fraction of doublets (5-20 per cent from electron microscope counting) by a computer simulation of the combined spectra of two component translational and rotational diffusion theory as presented by CUMM1NS et aL (1969). The signals were considerably weaker than DBP runs and as such the data show greater variation from the regression lines. Despite this variation the correlation coefficients are all greater than 0.94. The errors between the regression lines and the theoretical lines based on manufacturer's specified sizes range from 5.5 to 8.2 per cent and appears to be systematic. One possible explanation is that the particles were not completely dry or had a coating of impurities, and had an effective diameter greater than the manufacturer's. Perhaps a more likely explanation is that the effect of the doublets was underestimated, a likely result of CUMMI~Set aL (1969) theory in this situation of greatly extended Rayleigh-Debye conditions. It is a natural question to ask what is the effect of the laser beam on the aerosol particles and their motion. Under the conditions of these experiments the effects were negligible. The minimum beam diameter was 0-05 cm which gives a beam intensity of --~ 2.5 W/cm-" or about 16 times the intensity of unfocused sunlight. Because the particles were non-absorbing

500--400 N I

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--------Theoretlcai

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Regression hne ?- 0 557 ~m dia. & 0 796 #~m dla,

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F t 6 . 9 . Half-width versus K 2 for 0"557, 0-796, 1-10 tLm diameter polystyrene latex aerosols. Dashed lines are theoretical lines based on manufacturer's taarticle size and solid lines are regr~n~ion lines through the origin. Scattering angles used were 20 °, 25 °, 30 °, and 35 °, - - - Theoretical line; regression line © 0-557 ~,m dia.; A 0:796/~m dia.; × 1-10t~m dia.

Aerosol measurement by laser Doppler spectroscopy--I

513

heating was considered negligible compared to the temperature increases of 14 ° required to produce a 1 per cent change in the measured half width. Similarly we neglect photophoresis in agreement with AvY (1965). The radiation pressure is given by KERKER (1969) as, F = r r d 2 - IQp, 4c

(23)

Using a value of Qp, --~ 1 the radiation force on a 1 /zm particle would be ,-~ 7 × 10 -11 dynes which would give a maximum drift velocity of ,-- 5 × 10 -4 cm/sec, however, the particle is only in the beam for 0.025 seconds and can therefore move only ~/5 cm while in the beam, a negligible amount. Furthermore, for homogeneous aerosols all particles would move together which would produce no effect on the homodyne spectrum because it depends only on the relative motion of the particles. The 0 and A6 errors contributed less than a 2 per cent error to the measured half width. The effects of ambient dust particles in the laser beam were minimized by purging the light tight box with filtered air before each run. For the concentrations and angles used the intensity of multiple scattered light was negligible, less than 10 -a of the singly scattered light. The combined error of frequency and spectral distortion and noise is estimated to cause less than a 5 per cent error in half width. The primary source of error is the random error in the measured signal which is in an inherent property of the stochastic process being measured. For the total sampling times used of 300 to 700 seconds this error was generally less than 10 per cent except for the PSL measurements where because of the weak signal the errors of individual spectra ranged up to 30 per cent. This error decreases with sampling time being proportional to t -1/2. F r o m the results presented here we see that laser Doppler spectroscopy works satisfactorily as an aerosol sizing technique and that it is theoretically and experimentally equivalent to the liquid suspensions evaluated in earlier work. As can be concluded from the theory and results presented here, having the aerosol flow through the scattering plane does not affect the measured HW. Except for finite measurement time LDS has the advantages of conventional light scattering and the additional advantages that for monodisperse aerosols particle refractive index does not affect the results; the equipment does not require any absolute intensity calibration; and at small scattering angles LDS is insensitive to particle shape. Acknowledgement--This investigation was supported by Public Health Service Training Grant No. 8 TO1

OH 00016-I1 and by National Aeronautics and Space Administration Research Grant No. NGL-22-007-053.

REFERENCES

AREccm, F. T., GIGLIO,M. and TARTAR1,U. (1967) Phys. Rev. 163, 192. AvY, A. P. (1963) as quoted by CADLE,R. D. (1965) Particle Size. Reinhold Publishing Co. BENEDEK,G. B. (1969) Polarization, Matter, and Radiation. Presses Universitaire de France, Paris. CHU, B. (1970) A. Rev. phys. Chem. 21, 145. CUMMINS,H. Z., CARSON,F. D. HERBERT,T. J. and WOODS,G. (1969) Biophys. J. 9, 518. CUMMINS,H. Z., KNABLE,N. and YEI-I,Y. (1964) Phys. Rev. Letters 12, 150. CUMMINS,H. Z. and SWlNNEY,H. L. (1969) Progress in Optics. (Edited by WOLF, E.) 8. DUBIN, S. B., LUNACEK,J. H. and BENEDEK,G. B. (1967) Proc. Natn. Acad. Sci. U.S.A. 57, 1164. FORD, N. C. and BENEDEK,G. B. (1965) Phys. Rev. Letters 15, 649. FRENCH, M. J., ANGUS,J. C. and WALTON,A. G. (1969) Science, 163, 345. FUCHS, N. A. (1964) The Mechanics of Aerosols. Pergamon Press, Oxford. KERKER,M. (1969) The Scattering of Light. Academic Press, New York.

514

W. I-hNva and P. C. l~ls'r

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