Download Calibration, Size bin choices, Mie scattering theory

Size Calibration of the DMT Single Particle Light Scattering Probes
(SPP 100, 200, 300, FM, CAS and CAPS)
Darrel Baumgardner
Droplet Measurement Technologies
Boulder, CO
The operating principle of instruments that detect particles by the amount of light they
scatter when passing through a focused light beam is based on the simple concept that the
amount of light scattered by a particle is proportional to its size. The following discussion
explains how the light gathered in the collecting optics of the instruments is subsequently
converted to a size. Part of this discussion will explain where sources of uncertainty arise that
limit the resolution of these types of instruments
Briefly, this class of instruments, called optical particle counters (OPCs) from this point
on, use a laser to illuminate a volume in the air stream. As the particles pass through the laser
beam they scatter light. Some fraction of this is collected by the optics and focused on a
photodiode. The peak amplitude of each pulse is compared with user-programmed voltage
thresholds (10, 20, 30, or 40) and a count is assigned to the bin number that corresponds to the
channel numbers whose upper and lower thresholds bracket the pulse height value.
Since it is the scattered light that is being measured we need a way to relate this to the
particle size. We first assume that the particle is spherical and then select the particle refractive
index, the intensity and wavelength of the illumination, the angles over which the light is
collected, the efficiency of the detector and the electronic gain. With this information we can use
Mie scattering theory to calculate the relationship between pulse height measured and particle
size. We don’t precisely know the illumination and detector efficiency, so an empirical
calibration is required to assign sizes to each of the pulse height thresholds that have been set in
the instruments.
If we have calibration particles whose size we can strictly control, then we can sample
these particles with the probe and get a relationship between particle size and voltage threshold.
The ideal method would be to generate particles with the refractive index of the atmospheric
particles we plan to measure. We would start with the lowest size threshold and adust the size
until we only detect particle in the first channel. This would then define the size threshold of that
channel. Subsequent adjustments would then calibrate the higher bin sizes. Unfortunately, for
instruments that measure water, like the FSSP-100, generating such a precise size of droplet is
very difficult (but not impossible). For most routine calibrations, therefore, most researchers have
substituted less volatile particles for water and then made adjustments for the difference in
refractive index. For example, commercially available glass beads, with well controlled sizes, are
the normal calibration particle used. In principle, we should be able to select a range of glass
bead sizes, convert them to their water equivalent size (glass and water scatter light differently,
so we have to know what size water particle will scatter the same amount of light as a particular
glass bead size), and perform the threshold calibration as described above.
Two issues complicate this concept. One, the amount of scattered light is dependent on
the index of refraction of the particles, as well as their size. Two, in certain size ranges the
scattered light does not increase monotonically with size. Figure 1 illustrates these two points.
Figure 1
In this figure we see that glass particles (refractive iindex, n=1.51), in the size range
shown, scatter less light than water droplets (n = 1.33). Secondly, it can be seen that at some
sizes, particles scatter less light than smaller particles. This complication leads to a big problem
in trying to use glass beads as a surrogate for water. In the Fig. 1 example, a commercially
available glass bead with a size of 20.6 m, has a scattering cross section of approximately 5x107
cm-2. We can see that several sizes of water droplets scatter the same amount of light, i.e. 16.5,
17, and 18 m water droplet have approximately the same scattering cross sections (intersection
points of horizontal blue line). In addition, the size of the calibration beads are only controlled to
approximately ±1% , and inhomogeneities in the intensity of the laser beam add to the
uncertainty in measured light, so a conversion to water equivalency could produce a range of
possible water droplet sizes from approximately 15 - 19 m. This is too much uncertainty to
determine the voltage thresholds of the instruments. Therefore, we take a different approach that
side steps this problem.
If we believe the Mie scattering theory, and we do since it has been verified
experimentally in many laboratory studies, we can convert the voltage thresholds of the OPCs
with only a single point measurement. The procedure is as follows:
2
Step 1 - Measure precisely the preset voltage thresholds of the OPC, V1, V2, …, Vn.
Step 2 - Select a glass bead particle, D0, whose size is approximately in the mid-size range of the
instrument.
Step 3- Sample these beads with the OPC until a statistically reasonable number of particles
have been counted and a well defined peak is seen in the frequency histogram generated
on the data system interfaced to the instrument.
Step 4- Convert the average channel number where peak counts are seen into a voltage, V0. This
can be done by calculating either a geometric or arithmetric average of the voltages that
comprise the upper and lower threshold of the channel. The geometric average is
preferable since the voltage thresholds increase exponentially rather than linearly.
Step 5- Calculate the scattering cross section, I0 (Fig. 1) of the D0 calibration particle.
Step 6- Calculate the scale factor, SF, that relates volts to scattering cross section, i.e. SF = V0/I0.
Step 7- Convert each voltage threhold, Vi, to an equivalent scattering cross section, Ii , i.e. Ii =
SF*Vi.
Step 8- Find what size of water particle di has the scattering cross section, II by using a table of
calculated Mie scattering cross sections, and assign di to threshold Vi.
Again, in principle, this appears quite straight forward, but the multivalued nature of the
scattering cross section of particles complicates things. What size do you assign to a voltage
threshold when several sizes can produce the same amount of scattered light as was shown in
Fig. 1. One possibility is to adjust the voltage thresholds so that they bracket those multivalued
regions, e.g. , as shown in Figure 2.
Figure 2
3
The vertical dashed lines are selecting regions that insure that particles won't be
incorrectly placed in lower size bins. This technique is preferable if we know the refractive index
of the particles exactly. This is probably a good assumption when measuring cloud droplets, but
for aerosol probes like the SPP-200 and SPP-300, we usually don't know the particle refractive
index, and the shape of the Mie curve will change, depending on the particle composition.
Therefore, since as many as 40 size thresholds can be programmed into the DMT SPP
electronics, it is probably a better practice to cluster a number of channels around the
multivalued regions and set the thresholds so that they are relative close together. In subsequent
measurements, this type of set up will produce a size distribution that may look rather ragged, as
in the example shown in Fig. 3.
Measurements with Original Bins
Measurements after grouping bins
200
Concentration
150
100
50
0
2
4
6
8
10
12
14
16
18
20
Particle Diameter (um)
Figure 3
By having multiple bins, the spectra can be sensibly smoothed by combining the counts in
bins that adjoin the delinquent sections, i.e. where fluctuations in the spectra go against physical
reasoning. In fig. 3, bins 4-7 and 9-11 have been combined to produce an adjusted size spectra
with smoother shape.
Another alternative to the above two approaches, if the user is reluctant to set thresholds
whose size values are likely to be uneven in width (this will certainly be the case in either of the
above two approaches), is to assume that other uncertainties in the measurement process will
more or less smear out the effects of Mie oscillations. In this case, a monotonically increasing
curve can be fit through the Mie cross section values as shown in Fig. 4. Instead of adjusting
voltage thresholds to bracket the multivalued regions, the theoretical scattering cross sections
4
were smoothed with a series of power law fitsto produce a monotonically increasing function.
The voltage thresholds can then be set to values where the channels correspond to evenly spaced
sizes as shown in this figure.
Figure 4
It this technique that is used by PMS Inc. to set the fixed thresholds of the 15 size bins in
their FSSP-100 (without the DMT SPP electronics), as shown in Fig. 4 over a sub-set of this
instument's size range.
This method of setting the channels provides nice size intervals, but masks the fact that
there is a fair degree of uncertainty associated with these size intervals, e.g. a certain fraction of
particles in size range 11 - 14 m may actually get sized in the 14-17 m size range, while there
will be some particles in the 14-17 m size range that end up in the lower channel. This lends
uncertainty in the sizing of particles by the FSSP and subsequent calculations of the LWC that
depends on the 3rd power of the diameter. Since we can calculate these fractions of under and
over sizing in the instrument, it should be possible to do inversions on the spectra and obtain a
better estimate of thee sizes. There is no published work on this possibility.
At this time, even though there are possible uncertainty in using the smoothed curve
techniqe, DMT sets the size thresholds in new SPP units, or those calibrated for a user, to evenly
5
spaced intervals. This is done because there will indeed be some smearing of the Mie oscillations
that result from the inhomogeneities in the laser beam, response time roll-off, and other types of
noise that is inherent the system. This technique is also used so as to provide a consistent
methodology that is easily understanfable by the novice user. For the experienced user of the SPP
equipped probes, however, DMT recommends the technique of utilizing the maximum 40
channels and clustering the majority of channels in the region of maximum Mie fluctuations.
This region is over the size range 1 - 20 µm. Therefore, approximately 30 of the size channels
could be evenly spaced over this size interval, with the remaining 10 channels set at even
intervals from 20-50 µm.
6