Download Acoustic Measurement of Aerosol Particles

Amended version of article published:
Aerosol Science and Technology 19:339-350 (1993)
Acoustic Measurement of Aerosol Particles
Reagan Cole* and Kevin B. Tennal
Room 575 ETAS Building
2801 S. University Avenue
University of Arkansas at Little Rock
Little Rock, AR 72204
* Corresponding Author
email: [email protected]
(501) 569 8041
ABSTRACT
Measurements of the motion of aerosol particles in an acoustic field were made on particles covering two
orders of magnitude (0.3 µm to 30 µm) in diameter and one order of magnitude in density. The
measurements of both velocity magnitude and phase agree well with the theoretical model presented by
König in 1891. As a result, the diameter and the density of spherical aerosol particles can be determined
simultaneously from the measurement of the particle velocity. The phase and magnitude of the acoustic
velocity can be determined by indirect methods, allowing particle sizing to be performed without the use
of precision particles for calibration standards.
BACKGROUND
Acoustic sizing of aerosol particles involves subjecting a test aerosol to a controlled, locally intense
acoustic field and measuring the response of individual particles in the aerosol to the stimulus. Particle
size may then be computed if particle response is known as a function of size. This technique apparently
originated with Gucker and Doyle (1956). In a seminal paper these authors proposed an instrument that
used the combination of a linear gradient optical filter and a phototube to transduce the particle velocity
into an electrical signal. They suggested that the velocity magnitude might be processed by "a rapid
electronic circuit to obtain the combined count and size distribution for an ensemble of particles, while the
phase lag of the particles also could be studied with a rapid electronic phase meter." However, only
particle displacement results obtained by microphotography were reported.
Laser Doppler velocimetry (LDV), was developed in the decade following the Gucker and Doyle (1956)
paper and is a nearly ideal motion transducer for free particle experiments. The technique is non-contact,
stable and has excellent spatial resolution. LDV was first applied to acoustic particle sizing by Kirsch and
Mazumder (1975). The resulting instrument could perform automatically the type of measurement
originally proposed by Gucker and Doyle. Shortly after this, Sato, Kishimoto and Sasaki (1978a) added
digital signal processing (DSP) to the LDV-based acoustical sizing scheme and explored several
interesting aspects of the technique including the use of non-sinusoidal forcing functions (Sato, Kishimoto
and Sasaki (1978b)) and simultaneous determination of density and diameter (Sasaki, Sato and Oda
(1980)). Mazumder, Hood and Ware (1983) described a modification of the LDV-acoustic technique
using a combination of acoustic and electric fields to measure simultaneously the charge and size of single
particles. An instrument based on their work, the E-SPART analyzer (US Patent #4,663,714), is
currently marketed by Hosakawa Micron, International (Osaka, Japan). Hunik (1987) described a
particle characterization system using DSP and simultaneous acoustic and alternating electrical fields.
MODELS
A rigorous treatment of the motion of a particle in an acoustic wave is extremely complicated.
Fortunately, the problem of spherical particles undergoing rectilinear motion can be treated by a simple
analysis. The equation of motion for spherical particles in an acoustic wave is given by Fuchs (1964, p.
84) as
(1)
m dV
dt
3 m dU
2
dt
1 m dV
2
dt
9
2
m
4
Dp
2
9 m 2
4
Dp
2
1
g
g
dV
dt
2
Dp
dU
dt
2
V
U
p
,
where
and
V = Particle Velocity
U = Acoustic Velocity
m = Particle Mass
m' = Mass of Displaced Gas
g = Gas Density
= Acoustic Angular Frequency
Dp = Particle Diameter
= Viscosity.
According to Fuchs, the first term on the right hand side of the equation is "the reaction of the medium on
the particle due to the pressure gradient in the medium." The second term is "that portion of the reaction
of the medium that depends on the acceleration of the particle with respect to the medium." The third and
forth terms, attributed to Stokes, represent the viscous drag between the particle and the medium. Note
that the reactive forces are proportional to the cube of the diameter of the particle while the drag forces
are proportional to the square of the diameter of the particle.
The steady state solution to equation (1), attributed by Fuchs (1964, p.85) to Walter König who
published in 1891, is:
1
V/U
f2
a
tan
3
2
9/2
3 f
2
9/2
3
9/2
9/2
9/4
3
9/4
4
4
,
p
3/2 (f
1
f (1
where
3/2
2
Dp
)
1)
3/2
2
(1
9/2
and
f
g
)
2
,
3
9/2
2
p
3
g
9/4
4
.
- p is the phase of the motion of the particle relative to the phase of the acoustic field and
density of the particle. This result is hereafter referred to as the König model.
a
p
is the
A simpler expression obtained from Stokes' law (i.e., considering only non-inertial resistance and, hence,
keeping only the third term of the right hand side of equation (1)) is (Gucker and Doyle (1956)) V = U
[1 + j ]-1, where is the angular frequency of the acoustic wave and is the relaxation time of the
particle defined as = Dp2 p / 18 . Transformed to polar form this result is,
V = U [1 + (
)2]1/2;
p
=
a
- tan-1[
]. (3)
This result is hereafter referred to as the Stokes model. Differences between the Stoke and the König
model will be explained below.
A more rigorous approach to the problem of a sphere in a sound wave was taken by Temkin and Leung
(1976). These authors started from first principles and obtained a solution that includes not only the
displacement terms but also the scattering of sound by the particle. This additional effect is only
significant for particles whose size is comparable to the acoustic wavelength. Numerical evaluation of the
rather complicated solution derived by these authors shows that it closely follows the König model over
several orders of magnitude in particle diameter.
The effect of molecular slip is not explicitly included in the derivations of the above models. However,
since our LDV can acquire useful signals from small (0.33 µm) test particles we found it necessary to
modify the models to include slip. To do this, we followed the example of Gucker and Doyle and applied
the Cunningham slip correction factor to each term containing viscosity. This approach is not justified by
any rigorous means, but does seem to improve the goodness of fit of phase versus diameter for both
Stokes and König models for small particles (Dp < 1 µm).
Several properties of the two slip
modified models should be noted. The Stokes model allows only the aerodynamic diameter, Dp p½, or
an equivalent property to be computed from the particle response. Knowledge of and provides the
product Dp p½ and not the particle diameter directly. The solution to the equation of motion for a particle
in the König model is given in terms of two dimensionless quantities and f, which, in principle, permits
both the diameter and the density of the particle to be determined from the observed motion.
Figure 1 shows a graphical comparison of the two models in a Nichols plot where the phase of the
particle motion with respect to the acoustic field is plotted versus the logarithm of the velocity magnitude
ratio V/U. The models are evaluated at an acoustic frequency of 6250 Hz with the two independent
variables, density and aerodynamic diameter, allowed to vary over the indicated ranges. The Stokes
model is represented by the single upper curve because all particles of the same aerodynamic diameter
are predicted to have the same motion regardless of particle density. The König model is represented by
the family of curves with each curve in the set corresponding to a different particle density. By inspection
it is seen that the two models converge for small particles and for very high densities. In the König model,
phase has a local maximum the value of which is a function of particle density.
PREVIOUS WORK
Despite the amount of work performed in acoustical particle sizing no data sufficiently comprehensive to
distinguish between the predictions of the two models have been reported. Gucker and Doyle (1956)
tested the displacement part of a comprehensive theory similar to that of König, and found it to hold
within the limits of experimental error. Gucker and Doyle also stated that it is just as well to use the
Stokes model because it predicts similar response magnitude and because it is computationally much
simpler. Due to the limitations of their apparatus, they were unable to test the phase part of the theory.
Sasaki, Sato and Oda (1980) used a solution to a somewhat simplified version of equation (1) in their
work on density measurement. They reported results on only two aerosols one of which was in an
uncontrolled gas mixture. Kirsch and Mazumder (1975) and Hunik (1987) both used the Stokes model
as a basis for their computations.
APPARATUS, METHODS and MATERIALS
Acoustic Source
An abbreviated diagram of the experimental apparatus is shown in Figure 2. A localized acoustic field is
produced in the measurement volume by a pair of Motorola KSN-1038 piezo tweeters wired in
antiphase and positioned on opposite sides of the LDV sensing volume. Inverse conical horns attached
to the tweeters allow high acoustic velocity levels with modest input powers and improved clearance for
the optical sensor since the throats in stead of the mouths of the horns intrude into the measurement
chamber. An acoustic velocity of approximately 0.5 meters per second is produced in the measurement
volume for a drive frequency of 6250 Hz. Useful velocity levels are achieved over the frequency range 3
KHz to 26 KHz. Acoustic drive frequencies of 6250 Hz and 25 KHz -- set by a digitally controlled sine
wave generator -- were used for the measurements reported.
The acoustic transducers are located in a sealed chamber containing an inlet and outlet pipe, a thermistor
and a microphone. Particles under test are introduced to a holding vessel on top of the test chamber and
are pulled at low speed through the chamber and exhausted into a safety vent.
Optics
A 12 mW He-Ne laser and an acousto-optic modulator (Bragg cell) comprise the LDV transmitting
section. The Bragg cell acts as both a variable ratio beam splitter and a frequency shift device imposing a
bias frequency of 40 MHz between the two beams of the LDV. Anti-reflective coatings on all
transmissive components and dielectric mirrors allow an optical throughput of approximately 90% of the
laser power. The interference fringe spacing is set at 2.4 µm, so that particle motion along the sensing
vector produces a Doppler shift from 40 MHz of 412 KHz per meter per second. Light scattered in the
near forward direction is collected using an f/1.4, 55 mm camera lens and focused onto a Hamamatsu,
model #1617, photomultiplier tube (PMT). With the laser beams focused to produce a 100 µm beam
waist at the sensing volume, processable signals are obtained for single particles as small as 0.33 µm in
diameter.
Signal Processing and Data Reduction
LDV processors are generally optimized for fluid flow studies in which the object is to detect signals from
monodisperse seed particles and to reject all other signals. Signals originating from particle analysis
experiments will have a much broader range of charateristics. For instance, for a 10 2 range in particle
diameter at least a 104 range in light scattering magnitude is expected. The range of detected velocities is
expected to be similarly large. For this reason, the signal processor used in this study is optimized for a
wide dynamic range with respect to light scattering (80 dB) and detected velocity (60 dB).
Improvement in the scattering dynamic range is achieved by reducing the background light level and
logarithmically compressing the PMT signal. A wideband frequency discriminator allows high resolution
and good linearity over a range corresponding to 1 - 1000 mm/s. A block diagram of the signal
processing electronics is shown in Figure 3. The photocurrent from the PMT is converted to voltage and
separated into low pass and 40 MHz band pass channels. The separated signals are processed to give
the logarithm of the low frequency or pedestal component (PED), the logarithm of the high frequency
component (RF) and the measured Doppler frequency (VELOCITY). Further signal processing derives
trigger signals, synchronizes the trigger to the acoustic excitation and multiplexes the PED and RF signals
onto a single channel. These signals, SCATTERING (RF and PED), VELOCITY and TRIGGER, are
then recorded using a LeCroy 9400a digital sampling oscilloscope (DSO) operating in a burst mode.
The DSO is capable of logging up to 250 signal bursts per buffer. Since TRIGGER is synchronous with
the excitation, the phase of the excitation signal may be determined by examining the trigger time buffer
associated with each burst record. A desktop computer with an internal general purpose instrument bus
(IEEE 488) interface and a local area network card is used to log the DSO output into mass storage.
Subsequent data analysis is performed off line using a combination of network facilities and the desktop
computer.
Due to the intensity distribution of light forming the LDV sensing volume, the passage of a particle through
the volume produces a signal burst with rapidly changing characteristics. A high pass filter allows
separation of the information and noise components of the signal. The high pass or noise component is
rectified and detected to yield a measure of signal quality. A fixed threshold operating on the detected
noise serves to delineate the high signal to noise ratio portions of the velocity signal. This "signal quality"
detection scheme proves to be very reliable and robust. For instance, the particle detection rate is very
nearly independent of the PMT gain. As a result, only processable signals are logged and data storage
requirements are minimized.
A discrete Fourier transform is performed on the windowed portion of the VELOCITY signal from each
particle to extract the phase and the magnitude of the signal at the acoustic frequency. Velocity, scattering
magnitude and other observations for individual particles are then copied to a data base. A commercial
database program and supplementary routines written in a standard programming language are used for
additional analysis.
EXPERIMENTS and RESULTS
Signal Recovery
An example of the output of the analog LDV processor for a 0.5 µm particle in a 25 KHz acoustic field is
shown in
Figure 4. Three traces are shown for the particle burst representing the RF and PED outputs of the
processor and the VELOCITY signal.
With the burst is a listing of the reduced data identifying:
* Particle serial number,
* Relative phase, in radians, of the particle motion at the acoustic frequency,
* Velocity magnitude of the particle motion at the acoustic frequency,
* Mean velocity of the particle along the LDV sense vector,
* Average noise level during the analytical window,
* Log of the low frequency component of the scattering current or pedestal (PED),
* Log of the 40 MHz component of the scattering current (RF), and
* The duration of the windowed portion of the signal in milliseconds.
The magnitudes are given in units of volts which may be readily transformed to units of velocity or current.
The limits of the analytical window are denoted by vertical lines in the Figure 4.
A reference aerosol of known size and density was used either during, or before and after, each
experiment in order to estimate the acoustic velocity and to ensure that the acoustic field remained stable.
Particle velocities for the test aerosols were then normalized to the measured response of the reference
particle.
Precision Spheres
Solid, monodisperse spheres of polystyrene latex (PSL) with a specific gravity of 1.05 (Duke Scientific,
Inc., Palo Alto, CA) ere aerosolized from aqueous suspension by pneumatic nebulization and dried over
anhydrous silica gel. Several sizes in the range of 0.33 µm to 8.7 µm were used.
The relative phases and velocity magnitudes for aerosols consisting of five sizes of precision PSL spheres
are shown in Figure 5 for an acoustic frequency of 6250 Hz. The test aerosols were introduced
sequentially to the apparatus. Phase is plotted in radians and is referenced to the computed acoustic
excitation based on measurements for the 0.33 µm diameter aerosol. The relative velocity of the particle
motion is given on the horizontal axis where 1.0 represents the velocity magnitude for the excitation.
Histograms of the relative count versus phase and count versus velocity are shown in separate boxes
outside the main graph. The peaks for the 0.33 and 1.088 µm aerosols are not separated in the relative
velocity measurement but are clearly separated in phase. Figure 6 shows the modal values of the velocity
magnitude and the phase plotted against particle diameter. The theoretical predictions of the Stokes and
König models are also shown.
The same technique was used to obtain the responses of test particles at 25 KHz. Results are shown in
Figures 7 and 8 . The 8.7 µm PSL used in these tests had a 17.2% coefficient of variation as specified
by the manufacturer, compared with 0.6% to 2.5% for the smaller PSL test aerosols. The spread in
phase seen for the 8.7 µm size is predicted by the König model. Also apparent on the plots is the smaller
velocity magnitude observed for the 0.33 µm particles relative to slightly larger particles. This was found
to be due to the performance of the frequency discriminator under marginal signal conditions resulting
from low scattered light levels. Measurements on the 0.605 µm diameter aerosol were used to calculate
the acoustic velocity for this data set. Note that in Figures 6 and 8 the relative velocities are
adequately predicted by either of the two models. Measurements of phase, however, lie very nearly on
the König model curve which deviates significantly from the Stokes curve.
BES and Copper Powder
According to the König model, the motion of a particle in an acoustic field depends on both the particle
diameter and particle density in a distinguishable manner. To test this prediction, measurements made on
polydisperse aerosols of different densities were compared. Spherical copper powder (-325 mesh, Alfa
Products, Danvers, MA) with a specific gravity of 8.96 was aerosolized by blowing it off a spatula blade
with a pulse of compressed air. A polydisperse aerosol with a specific gravity of 0.912 was produced by
spraying a low vapor pressure oil (bis-Ethylhexyl Sebacate (BES), Pfaltz & Bauer, Inc., Waterbury, CT)
from a manually operated, pump type, "airless" sprayer. A 2.08 µm diameter PSL aerosol was used for a
reference during each of the tests. The results are shown in Figure 9. The responses of the BES and
copper aerosols clearly fall on different curves. The entire ensemble of BES and copper particle
responses was collected in a database and normalized to the response of the reference particle. Then a
numerical technique was used to solve simultaneously the König equations to yield the aerodynamic
diameter and specific gravity of the individual particles. These solutions are plotted in Figure 10 and are
in accordance with the predictions of the König model. However, the effect of particle density on the
motion is too weak to serve as a basis for measuring the densities of particles with small aerodynamic
diameter. For instance, at one µm aerodynamic diameter, a ten milliradian shift in phase maps into a one
hundred to one change in density. For this reason, experimental measurements for particles with
aerodynamic diameters less than about three µm (at 6250 Hz) cannot not be used to determine density.
In the case of BES droplets, the mean specific gravity (sg) for particles with a computed diameter greater
than three µm is very close to the bulk value for BES (sg = 0.912). The results for copper spheroids in
the same range gives a mean specific gravity very near to the bulk density of copper (sg = 8.96). The
spread in apparent density, however, is much larger for the copper particles than for the BES droplets.
Examination of the copper sample using SEM showed the particles to have a wide range of aspect ratios.
This indicates that shape also influences the response.
Large Monodisperse Droplets
To illustrate the particle size resolution of the complex velocity or combined phase-velocity magnitude
technique, a series of measurements were made on large monodisperse droplets of BES produced with a
Vibrating Orifice Aerosol Generator (VOAG) (TSI, Inc., St. Paul, MN; model 3450). The relative
responses are shown in Figure 11 along with the calculated nominal sizes for the droplets. A PSL
reference aerosol (3.09 µm diameter) was used with each run, but the theoretical phase and relative
velocity for the reference could not be determined due to unknown gas parameters resulting from a high
concentration of alcohol vapor in the air. However, the data illustrate the high resolution of the instrument
for particle size separation and clearly show the decreasing phase lag with increasing particle diameter for
large particle diameters as predicted by the König model.
DISCUSSION
In order to determine the diameter and density of particles by measuring their motion in a driving acoustic
field, the phase and velocity of the acoustic field must be known. We are unaware of any direct, noninvasive means of measuring acoustic velocity. Some experimenters have attempted to measure the
acoustic velocity using a microphone. In a small reflective cavity near a real output transducer there is no
straightforward relationship between acoustic pressure as sensed by a microphone and acoustic velocity
as experienced by the particle. Only in a pure traveling wave does a simple proportionality exist between
the two quantities. Therefore, some alternate method to direct measurement must be used. In one
method measurements are made on particles sufficiently small that they can be assumed to follow the
motion of the gas very closely. The phase and velocity of the particle are then taken to be those of the
acoustic field. Larger monodisperse particles that do not follow exactly the motion of the gas may be
used providing their size and density are known and the correct theoretical response of the particles to the
gas motion is known.
In the König model there is a phase maximum that is a function of density. This feature allows a
heterodisperse aerosol of spherical particles of known density to be used for calibration. The phase of
the acoustic field is estimated based on the maximum phase shift observed for the aerosol. The maximum
observed velocity in this case is a reasonable estimate of the acoustic velocity.
A more refined calibration can be made using the heterodisperse aerosol by inverting the König model to
compute particle density from the measured phase and velocity and then adjusting the reference phase
and velocity until the computed mean density of all the particles is equal to the bulk density of the material
from which they are formed. Due to the insensitivity of the model to particle density at small particle
sizes, only particles with velocities less than about half the maximum observed velocity should be used in
the calculation.
It is apparent from Figure 10 that within a limited range the density of spherical aerosol particles can be
determined from their acoustic response. If density measurement is a primary objective, then acoustic
measurements could be made in a gas of a greater density than air. Measurement of the density of nonspherical particles poses a more difficult problem. With more analytical work, it may be possible to
identify a meaningful shape factor from acoustically forced motion experiments. We have noticed that the
phase and magnitude of doublets of PSL spheres do not fall on the curve corresponding to spheres of unit
density.
The scope of these measurements may be extended by modeling the particle response in terms of transfer
functions. The velocity of the particle is then represented as the product of a potential velocity (in this
case the acoustic velocity) and a dimensionless, complex transfer function. By extension, this operational
calculus allows modelling of the response of a particle to multiple forces. Then the particle response is
given as the complex sum of the various potential velocities and their associated transfer functions.
Forces such as those resulting from non-sinusoidal acoustic waves, magnetic and electrical fields, etc. can
all be treated simultaneously using this approach. If these forces act on orthogonal vectors, or are
orthogonal in time, the response of the particle to each force can be isolated and analyzed independently.
The calibration of such instruments is possible from first principles. For instance, a combination of
electrical and acoustic forces could be used to determine the acoustic velocity of the gas. This can be
done if the phase of the electrical field is accurately known by making use of the functional mapping:
a( e) = P( e) where P( e) is an approximating polynomial. Since the phase of the electrical field can be
directly measured, the phase of the acoustic field can be accurately determined by observing the motion
of a charged particle.
Although this study deals only with measurements of the velocity of particles, it should be mentioned that
the recorded scattering signals also contain particle size information and that the scattering magnitude and
the fringe visibility are easily computed from recorded signals.
CONCLUSION
The acoustic response of aerosol particles closely follows the König model cited by Fuchs (1964, p.85)
over a wide range of frequencies, densities and particle sizes. This response may be measured using
optical velocimetry and inverted numerically to determine the aerodynamic diameter and, in some cases,
the density of particles in an aerosol sample. In this scheme, sizing may proceed from first principles
without the use of calibration standards if precautions are taken to insure the stability of the acoustic field.
We conclude that it is possible to use the phase and magnitude of the acoustic response of aerosol
particles to secure particle size measurements over approximately 2 orders of magnitude with monotonic
excitation. At low ultrasonic frequencies, this would cover the range of approximately 0.3 µm to 30 µm
We also conclude that the technique allows the measurement of the density of spherical particles over a
limited range.
A patent is pending* (US Serial No. 07/956,296) on some aspects of the measurement technique and a
more detailed publication describing the technique is in preparation.
*US Patent #5,296,910, Mar. 22, 1994, 29 claims
9
REFERENCES:
Fuchs, N. A. (1964), The Mechanics of Aerosols
, Dover, New York.
Gucker, F. T. and Doyle, G. J. (1956), J. Phys. Chem.60:989-996.
Hunik, R. (1987), Kema Scientific & Technical Reports
5(4):73-81.
Kirsch, K. J. and Mazumder, M. K. (1975), Appl. Phys. Lett.26(4):193-195.
Mazumder, M. K., Ware, R. E. and Hood, W. G. (1983). In Measurements of suspended particles
by quasi-elastic light scattering
(B. Dahneke, ed.), Wiley Interscience, New York.
Sato, T., Kishimoto, T. and Sasaki, K. (1978a), Appl. Optics17(2):230-234.
Sato, T., Kishimoto, T. and Sasaki, K. (1978b), Appl. Optics17(4):667-670.
Sasaki, O., Sato, T. and Oda, T. (1980), Appl. Optics9(15):2565-2568.
Temkin, S. and Leung, C. M. (1976), J. Sound and Vibration
49:75-92.
10
LDV Receiver
Audio Drive
6250 Hz OR 25 KHz
Power
Amplifier
Speaker
#1
Speaker
#2
LDV Transmitter
Sine Shaper
Timebase
Transducers
Synchronizer
Aquisition Trigger
Trigger
Generator
LDV Optics
Frequency
Discriminator
Channel 1 Digitizer
PM Tube
I to V
Converter
40MHz
Band Pass
Logarithmic
Detector
Multiplexer
HV Supply
1 MHz
Low Pass
Logarithmic
Detector
Channel 2 Digitizer