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This document is downloaded from DR‑NTU (https://dr.ntu.edu.sg)Nanyang Technological University, Singapore.
Particle Classification by the Tandem DifferentialMobility Analyzer–Particle Mass Analyzer System
Kuwata, Mikinori
2015
Kuwata, M. (2015). Particle Classification by the Tandem Differential MobilityAnalyzer–Particle Mass Analyzer System. Aerosol Science and Technology, 49(7), 508‑520.
https://hdl.handle.net/10356/79327
https://doi.org/10.1080/02786826.2015.1045058
© 2015 American Association for Aerosol Research. This is the author created version of awork that has been peer reviewed and accepted for publication in Aerosol Science andTechnology, published by Taylor & Francis on behalf of American Association for AerosolResearch. It incorporates referee’s comments but changes resulting from the publishingprocess, such as copyediting, structural formatting, may not be reflected in this document. The published version is available at: [http://dx.doi.org/10.1080/02786826.2015.1045058].
Downloaded on 31 Jul 2021 02:44:22 SGT
1
Particle Classification by the Tandem Differential Mobility Analyzer –
Particle Mass Analyzer System
by
Mikinori Kuwata*
Division of Earth Science and Earth Observatory of Singapore,
Nanyang Technological University, Singapore
E-mail: [email protected]
Submitted: October 21, 2015April 5, 2015
Submitted to
Aerosol Science and Technology
*To Whom Correspondence Should be Addressed
2
Abstract 1
Particle mass analyzers, such as the aerosol particle mass analyzer (APM) and the 2
Couette centrifugal particle mass analyzer (CPMA), are frequently combined with a differential 3
mobility analyzer (DMA) to measure particle mass mp and effective density ρeff distributions of 4
particles with a specific electrical mobility diameter dm. Combinations of these instruments, 5
which are referred as the DMA-APM or DMA-CPMA system, are also used to quantify the 6
fractal dimension Df of non-spherical particles, as well as to eliminate multiply charged particles. 7
This study investigates the transfer functions of these setups, focusing especially on the 8
DMA-APM system. The transfer function of the DMA-APM system was derived by multiplying 9
the transfer functions of the DMA and APM. The APM transfer function can be calculated using 10
either the uniform or parabolic flow models. The uniform flow model provides an analytical 11
function, while the parabolic flow model is more accurate. The resulting DMA-APM transfer 12
functions were plotted on log(mp)- log(dp) space. A theoretical analysis of the DMA-APM 13
transfer function demonstrated that the resolution of the setup is maintained when the rotation 14
speed ω of the APM is scanned to measure distribution. In addition, an equation was derived to 15
numerically calculate the minimum values of the APM resolution parameter λc for eliminating 16
multiply charged particles. 17
18
3
1. Introduction 19
Particle classification is a key technique for investigating aerosol particles (Hinds 1999; 20
McMurry 2000). Particle classification instruments, such as the differential mobility analyzer 21
(DMA), have been widely employed throughout all areas of aerosol research (Knutson and 22
Whitby 1975; Stolzenburg and McMurry 2008). Most of these instruments, including the DMA, 23
classify particles based on diameter dp, using the dynamics of the particles as classification 24
principles (Hinds 1999). 25
The particle mass analyzer (PMA), which includes both the aerosol particle mass 26
analyzer (APM) and Couette centrifugal particle mass analyzer (CPMA), is becoming a popular 27
tool to classify particle mass (Ehara et al. 1996; Olfert 2005; Tajima et al. 2011). The concept of 28
the APM was firstly introduced by Ehara et al. (1996), and the CPMA was proposed by Olfert 29
and Collings (2005). The PMA consists of two rotating cylinders, and a voltage is applied in 30
between them. This design allows the PMA to classify particles based on the balance between 31
the centrifugal and electrostatic forces. Since centrifugal force is proportional to particle mass mp, 32
the PMA is capable of classifying particles based on their mass. In the case of the APM, two 33
cylinders rotate at the same angular velocity for accurate mass classification (Ehara et al. 1996). 34
On the other hand, the rotation speeds of the two cylinders are different for the CPMA, which 35
allows the instrument to have a higher particle transmission than the APM (Olfert 2005). 36
In many cases, the PMA is combined with the DMA in tandem (McMurry et al. 2002; 37
Kuwata et al. 2009; Cross et al. 2010). Examples of these setups include the DMA-APM, 38
DMA-CPMA, and APM-scanning mobility particle sizer (SMPS) systems (McMurry et al. 2002; 39
4
Malloy et al. 2009; Cross et al. 2010). In these setups, particles are classified by both electrical 40
mobility diameter dm and mp, which allows for the quantification of important physical 41
parameters, such as effective density ρeff, dynamic shape factor, and mass-mobility exponent Df 42
(Park et al. 2003; Kuwata and Kondo 2009; Zangmeister et al. 2014). The combination of these 43
two techniques is useful in eliminating multiply charged particles because the classification 44
regions for multiple charged particles of the DMA and PMA do not overlap (Pagels et al. 2009; 45
Shiraiwa et al. 2010). 46
However, the instrumental responses of these setups, which are useful in optimizing 47
experimental conditions, have not been evaluated theoretically. This study develops the transfer 48
functions of the DMA-PMA setup by focusing on the APM. Implications of the theoretically 49
derived transfer functions on actual operation will also be discussed. 50
51
2. Mass-mobility relationship 52
2.1 Effective density and mass-mobility exponent 53
The relationship between mp, dm, and ρeff is shown by the following equation (McMurry 54
et al. 2002; DeCarlo et al. 2004). 55
31
6p e ff m
m d (1) 56
The equation is rewritten as follows in the logarithmic scale 57
1
lo g lo g lo g 3 lo g6
p e ff mm d
(2). 58
Equation 2 has the advantage of considering particle classification by both mp and dm since the 59
relationship is linear in the log(dm)- log(mp) space. This equation can be equally applied to both 60
5
spherical and non-spherical particles because ρeff depends both on the material density and 81
morphology of the particles (Park et al. 2003; Kuwata and Kondo 2009). Figure 11a plots the 82
relationship between mp, dm, and ρeff in the log(dm)- log(mp) space. This space is convenient for 83
deriving the DMA-PMA transfer function because the DMA and PMA classify particles by dm 84
and mp, respectively. 85
The log(dm)- log(mp) relationship can also be represented using other metrics, such as 86
the mass-mobility exponent (Df), which is calculated by the following equation (DeCarlo et al. 87
2004; Cross et al. 2010; Sorensen 2011; Zangmeister et al. 2014). 88
lo g lo g lo gf
D
p f m p f f mm d m D d (3) 89
Completely spherical particles have Df = 3, while the value is smaller for non-spherical particles. 90
Although the definition of Df is similar to that of the fractal dimension, these two parameters are 91
not equivalent (Sorensen 2011). Examples of log(dm)- log(mp) relationships for different values 92
of Df are shown in figure 11b. As indicated by the logarithmic form of equation 3, Df 93
corresponds to the value of the slope in the log(dm)- log(mp) space. This metric is especially 94
useful for characterizing the structure of aggregate particles, such as soot (Park et al. 2003; 95
DeCarlo et al. 2004; Zangmeister et al. 2014). The values of ρf and 1/6πρeff are equivalent when 96
Df is equal to three (equations 2 and 3), meaning that equation 3 may be considered as a 97
generalized form of equations 2. 98
99
2.2 Particle population on the log(dm)- log(mp) space 100
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Particles can populate on the log(dm) –log(mp) space in different ways, depending on 101
their morphology and mixing state. Three different types of particle populations are considered 102
here, namely spherical (or nearly spherical) particles with a constant value of ρeff, aggregated 103
non-spherical particles with a certain value of Df, and a mixture of spherical and non-spherical 104
particles with a range of ρeff (Figure 22). 105
Examples of spherical/nearly spherical particles with a constant value of ρeff include oil 106
droplets, ammonium sulfate, and sodium chloride (Kuwata and Kondo 2009; Tajima et al. 2011; 107
Tajima et al. 2013). In these cases, particles populate only on a line in the log(dm) –log(mp) space, 108
which has an intercept of 1
lo g lo g6
e ff
and a slope of three (equation 2). The intercept 109
is dependent on both the particle morphology and chemical composition, since these parameters 110
determine ρeff. An example for this case is shown in figure 22a, in which ρeff is assumed to be 111
1000 kg m-3. In this case, the particles can only populate on the black solid line in the figure. 112
Figure 22b presents an example of the second case, which corresponds to a constant 113
value of Df. As discussed in section 2.12.1, the value of the slope in the log(dm) –log(mp) space is 114
smaller than three for aggregated particles, such as soot (Park et al. 2003; Cross et al. 2010; 115
Zangmeister et al. 2014). In figure 22b, a mass-mobility relationship measured by Park et al. 116
(2003) is shown as an example. The particles populate only on the black solid line. The line is 117
not parallel to the isodensity lines because the mass-mobility exponent is smaller than three. As a 118
result, ρeff is smaller for larger particles (Park et al. 2003). 119
Figure 22c illustrates an example of an area for particle population for a mixture of 120
spherical and non-spherical particles with a range of ρeff (i.e., external mixture of various types of 121
7
particles). For example, ρeff of atmospheric sub-micron particles can have broad distributions 122
because many different types of particles, such as non-spherical soot particles, primary organic 123
aerosol particles, and secondary particles exist in the atmosphere (McMurry et al. 2002; Kuwata 124
and Kondo 2009). The upper limit of ρeff is determined by the material density of the heaviest 125
compound in the particles, and the lower limit of ρeff depends on both the material density of the 126
lightest species and the particle morphology. 127
128
3. Theory 129
3.1. Differential mobility analyzer (DMA) transfer function 130
The DMA classifies particles based on electrical mobility ,p m qZ d , which is defined by 131
the following equation (Knutson and Whitby 1975; Stolzenburg and McMurry 2008) 132
,
,
,3
c m q
p m q
m q
q e C dZ d
d (4) 133
where q is the particle charge, e is the elemental charge, and μ is the viscosity of a fluid (air). The 134
suffix dm (i.e., q) indicates the number of charges on a particle. Cc(dm,q) is the slip correction 135
factor, which is calculated using the mean free path of air l as 136
, , ,1 2 / 1 .1 4 2 0 .5 5 8 ex p 0 .9 9 9 / 2
c m q m q m qC d l d d l
(Allen and Raabe 1985). In a 137
certain DMA operating condition, the mode mobility of the classified particles *
pZ is calculated 138
as (Knutson and Whitby 1975; Stolzenburg and McMurry 2008) 139
2 _ 1 _*ln /
2p
sh D M A D M A
D M A D M A
Q r rZ
V L (5). 140
8
In this equation, r1_DMA and r2_DMA denote the inner and outer radii of the DMA, and LDMA is the 160
length of the DMA. VDMA stands for the DMA voltage, and Qsh is the sheath flow rate. Qsh is 161
typically controlled as equal to the excess flow rate of DMA (Wiedensohler et al. 2012). This 162
condition is assumed when deriving equation 5 and is employed throughout this study. 163
The DMA transfer function (Ω) is calculated by the following equation when particle 164
diffusion is negligible (Knutson and Whitby 1975; Stolzenburg and McMurry 2008). 165
, , ,
1, 1 1 2 1
2p p p p
m q m q m qZ Z d Z d Z d
(6) 166
In equation 6, *
, ,/p
m q p m q pZ d Z d Z and β represent the ratio of the sample and the sheath 167
flow rates. An example of the non-diffusing DMA transfer function is shown in figure 33a. 168
Although Ω is symmetric in the electrical mobility space, the shape of the function is skewed in 169
the diameter space because of Cc(dm). The minimum, central, and maximum electrical mobility 170
diameters for particle classification are denoted as dmin,q, dc,q and dmax,q (figure 33a). These values 171
are calculated by the following equations 172
m in ,1p
qZ d (7) 173
,1p
c qZ d (8) 174
m ax ,1p
qZ d (9). 175
As shown in figure 33a, Ω is separated into three regions by dmin,q, dc,q and dmax,q. In regions 1 (dm 176
< dmin,q) and 3 (dm > dmax,q), no particles are classified. The particles in region 3 (dmin,q ≤ dm ≤ 177
dmax,q) can pass through the DMA. 178
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179
3.2. APM transfer function 180
This section briefly introduces the transfer function of the APM, which was derived by 181
Ehara et al. (1996). The APM transfer function could be considered a special case of the CPMA 182
transfer function, as discussed by Olfert (2005). The APM transfer function has an analytical 183
solution, which facilitates the theoretical analysis of the DMA-APM response (section 3.33.3). 184
The APM classifies particles based on the balance between the centrifugal and 185
electrostatic forces, which is expressed by the following equation 186
,
2 2
_ 2 _ 1 _
2
, 2
_ 2 _ 1 _
ln /
ln /
c q A P M
c
c A P M A P M A P M
c q A P M
c A P M A P M A P M
m Vs
q e r r r
q em V
r r r
(10). 187
Specific mass s, which is calculated as m/qe, is a useful parameter for deriving the APM transfer 188
function. Suffix c indicates the central values of m and s of particles classified by the APM, and 189
suffix q corresponds to the number of particle charges. r1_APM, rc_APM, and r2_APM denote the inner, 190
center, and outer radii of the APM operating space, respectively. VAPM and ω are the voltage and 191
rotation speed of the APM. Equation 10 shows that both ω and VAPM can be adjusted to select sc 192
or mc. Ehara et al. (1996) has further demonstrated that the classification performance parameter 193
λ of the APM, which is defined by equation 11, plays a critical role in determining the APM 194
transfer function (Tajima et al. 2011). 195
2 2 2
2
, 2 _ 1 _22 p p m q A P M A P M A P M
A P M
A P M
m Z d L r rL
q eQv
(11). 196
10
This parameter is calculated as a function of relaxation time τ, ω, length of the APM operating 217
space LAPM, and the average flow velocity v . λ depends on mp, Zp, and the APM flow rate QAPM, 218
since τ and v are calculated as , , ,/ 3 /
p c m q m q p p m qm C d d m Z d q e (Seinfeld and 219
Pandis 2006) and 2 2
2 _ 1 _/
A P M A P M A P Mv Q r r , respectively. λ calculated for mc is 220
specifically named λc. The APM transfer function is conserved for a specific value of λc when it 221
is plotted as a function of normalized specific mass (s*/s) (Ehara et al. 1996). 222
The APM transfer function can be calculated either by the uniform or parabolic flow 223
model (Ehara et al. 1996). The uniform flow model has an analytical solution, which is 224
advantageous in theoretical analyses. On the other hand, the parabolic flow model provides a 225
more accurate form of the APM transfer function. Figure 33b shows APM transfer functions that 226
were calculated using these two models. 227
228
Uniform flow model 229
The uniform flow APM transfer function is separated into five regions by four 230
parameters (1
m
and 2
m
). The uniform flow APM transfer function has a maximum value of 231
e x pc
at 2 2p
m m m (region C). It monotonically increases/decreases in the regions of 232
1 2pm m m
(region B)/
2 1pm m m
(region D), respectively. The transfer function is zero 233
for 1p
m m
(region A) and1 p
m m (Region E). Table 11 summarizes the functional form for 234
the APM transfer function. 235
The values of 1
m
and 2
m
are calculated using the following equations. 236
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11
2 ,2
2
,_
2 2 , , _
1
1 /
ln ln ln ln 2 ln 1 /
q
c c qc A P M
c q c q c A P M
ms
s m r
s s m m r
(12) 237
1,1
2
,_
1 1, , _
1
1 / c o th / 2
ln ln ln ln 2 ln 1 / c o th / 2
q
c c qc A P M c
c q c q c A P M c
ms
s m r
s s m m r
(13) 238
In these equations, δ is calculated as 2 _ 1 _/ 2
A P M A P Mr r . The ratios of
2 , ,/
q c qm m
are 239
determined by the instrumental design, and the 1 , ,
/q c q
m m
ratios depend both on the 240
instrumental design and operating conditions, which is characterized by λc. Neither the APM 241
transfer function nor the mass resolution ratio changed for a specific instrumental design as long 242
as λc is conserved. 243
Figure 4 plots the values of λc, 1m
and
2m
in the log (mp)- log (dp) space. The value 244
of λc is smaller for larger particles because Zp is smaller (equation 11). This diameter dependence 245
leads to a broader APM resolution for larger particles, which also affects the DMA-APM transfer 246
function. 247
248
Parabolic flow model 249
A detailed description of the parabolic flow model APM transfer function is provided in 250
the Supplemental Information. For this model, 2 , q
m are calculated using equation 12, while 251
12
numerical calculations are required to obtain 1 , q
m . Numerical computation is also needed to 252
acquire the APM transfer function using the parabolic flow model. 253
254
3.3. Transfer function of the DMA-APM system 255
The transfer function of the tandem DMA-APM system (Φ) is calculated by overlaying 256
the transfer functions of both the DMA and APM (Radney et al. 2013). 257
, , , , , , ,p q m q p q m q m qm d m d d (14) 258
This equation can also be employed for the APM-DMA system because , , , ,p q m q m qm d d 259
is equivalent to , , , ,m q p q m qd m d
(Malloy et al. 2009). The properties of this equation are 260
examined in the following sections. The DMA-APM transfer function can be calculated for 261
seven different regions in the log(dm)- log(mp) space, as shown in figure 55 and table 33. 262
263
Region 1 (m m in
d < d ) 264
This region corresponds to region 1 in the DMA transfer function, meaning that no 265
particles in this region can pass through the DMA-APM system (i.e., , , ,0
p q m qm d ). 266
267
Region 2 ( m in m m a x
d d d ) 268
The particles in this size range are classified by the DMA. Particle transmittance in this 269
region depends both on the DMA and APM transfer functions. 270
13
Region 2A (m in m axm
d d d ,1p
m m
) 271
This region corresponds to region A in the APM transfer function, meaning that no 272
particles in this range can pass through the APM. 273
274
Region 2B (m in m axm
d d d , 1 2pm m m
) 275
This range of mp conforms to region B of the APM transfer function. Since both the 276
DMA and APM transfer functions are positive in this range, the DMA-APM transfer function is 277
positive in this region. 278
279
Region 2C (m in m axm
d d d , 2 2pm m m
) 280
In this area, region C of the APM transfer function overlaps with region 2 in the DMA 281
transfer function. Region C has the highest particle transmittance in the APM transfer function, 282
meaning the DMA-APM transfer function has the highest value in this region. The maximum 283
value is found at {dm, mp}={dc, mc}. The corresponding value of Φ is e x pc
when the 284
uniform flow model is employed to calculate the APM transfer function. 285
286
Region 2D (m in m axm
d d d , 2 1pm m m
) 287
The particles in this region pass through the APM, meaning that the DMA-APM transfer 288
function is positive. 289
290
14
Region 2E (m in m axm
d d d ,1 p
m m ) 311
The particles in this region cannot pass through the APM. Therefore, the DMA-APM 312
transfer function is zero in this region. 313
314
Region 3 (m m a x
d < d ) 315
This region corresponds to region 3 in the DMA transfer function. No particles in this 316
region can pass through the DMA-APM system (i.e., , , ,0
p q m qm d ). 317
Examples of the DMA-APM transfer functions are shown in figure 66. An example of 318
the uniform flow model for the APM is shown in figure 66a, and figure 66b demonstrates a result 319
for the parabolic flow model. These two transfer functions calculated using two different models 320
resemble each other, since the APM transfer functions for the corresponding operating 321
conditions are similar (figure 33b). In the following section, the characteristics of the 322
DMA-APM transfer function are mainly analyzed using the uniform flow APM model because 323
the analytical solution for the model facilitates detailed analyses. 324
325
3.4. Resolution of the DMA-APM system 326
The DMA-APM transfer function is surrounded by four points, which are denoted as P1 327
~P4 in figure 66. These points are located at P1{dmin, 1m
}, P2{dmax, 1
m
}, P3 {dmin, 1m
}, and 328
P4{dmax, 1m
}. The maximum mass of the classified particle mmax is observed at P2, while P4 329
corresponds to the minimum value of particle mass mmin. Both of those two points are located at 330
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15
dmax because λc is smaller for larger particles, which have smaller electrical mobility (equation 331
11). The masses at these points are calculated by equation 13, using dmax in calculating λc. 332
m a x ,
m a x ,
m a x ,
2
,_ ,
m in ,
2
,_ ,
1
1 / c o th / 2
1
1 / c o th / 2
q
q
q
c qc A P M c d
q
c qc A P M c d
m
mr
m
mr
(15). 333
This equation demonstrates that the mass resolution of the DMA-APM system is determined by 334
the instrumental design of the APM and λc at dmax. 335
Points P1 and P4 correspond to the minimum and maximum values of ρeff (ρeff_min,q, and 336
ρeff_max,q) 337
m a x ,
m a x ,
1, ,
_ m in , 23 3
m ax , m ax ,_ ,
6 6 1
1 / c o th / 2
q
q
d c q
e ff q
q qc A P M c d
m m
d dr
338
m in ,
m in ,
1, ,
_ m ax , 23 3
m in , m in ,_ ,
6 6 1
1 / c o th / 2
q
q
d c q
e ff q
q qc A P M c d
m m
d dr
(16) 339
These equations are rewritten as 340
m a x ,m in ,
m a x , m in ,
2
_ ,_ ,_ m a x ,
_ m in , _ , _ ,
1 / c o th / 2
1 / c o th / 2
q q
c A P M c de ff c de ff q
e ff q e ff c d c A P M c d
r
r
(17). 341
In this equation,,
_ ,p q
e f f c d corresponds to the ρeff of the particles with {dp, mp} = {dp,q, mc,q}. 342
These equations demonstrate that the density resolution of the DMA-APM system is determined 343
by both the DMA and APM resolutions. The density ratio of points A and B in figure 66, which 344
16
is calculated asm in , m ax ,
_ , _ ,/
q qe ff c d e ff c d
, corresponds to the density resolution derived solely from 345
the DMA resolution. The rest of the term in equation 17 346
m ax , m in ,
2
_ , _ ,1 / c o th / 2 / 1 / c o th / 2
q qc A P M c d c A P M c d
r r
matches the APM 347
resolution. 348
349
3.5. Apparent diameter resolution of the DMA-APM system 350
The mp resolution of the APM can be converted to dm resolution when the particles have 351
a uniform mass-mobility relationship (i.e., figures 22a and 22b), since mp can be converted easily 352
into dm when the relationship between these two parameters is uniquely known. In such cases, 353
the values of dm, which correspond to the minimum (dAPMmin) and maximum (dAPMmax) values of 354
mp classified by the APM, can be calculated using equation 3 as 355
m in
1
2 ,
m in
f
A P M
D
d
A P M
f
md
(18) 356
and 357
m a x
1
2 ,
m a x
f
A P M
D
d
A P M
f
md
(19). 358
Depending on the design and operating condition of the APM, dAPMmin may be larger than dmin, 359
and dAPMmax may be smaller than dmax. In this case, the apparent diameter resolution of the 360
DMA-APM system is determined by the APM rather than the DMA. 361
17
m a x
m in
1
1 ,m a x m a x
m in m in 1 ,
f
A P M
A P M
D
dA P M
A P M d
md d
d d m
(20). 362
Figure 77 provides an example. The particles populate on regions 2A and 2E in figure 363
77 (figure 55). In these regions, the particles cannot pass through the DMA-APM system even 364
though the DMA selects them because these areas are located outside of the APM classification 365
region. This situation occurs when the slope of the line connecting P2 and P3 is smaller than Df 366
(figure 77). This condition is written as 367
m a x m in
m a x
m in
2 , 2 ,
m a x m in
1
1 ,m a x
m in 1 ,
lo g lo g
lo g lo g
f
d d
f
D
d
d
m m
Dd d
o r
md
d m
(21). 368
It should be noted that even when the dm resolution of the DMA-APM system appears to be 369
controlled by the APM, the area for particle classification by the DMA-APM in the log(dm)- 370
log(mp) at a certain operating condition is still regulated by the DMA and APM (figure 77). 371
Although the apparent dm resolution of the DMA-APM system can be higher than the DMA 372
resolution, the actual dm resolution of the DMA-APM system is still controlled by the DMA. 373
When the particles have a broad distribution in the log(dm)- log(mp) space (figure 22c), 374
the diameter resolution of the DMA-APM system is predominantly determined by the DMA 375
resolution (dmin and dmax) because the particles distribute across the entire areas of 2A~2E. 376
377
378
18
4. Implication for instrumental operation 400
4.1. Operating the DMA-APM to investigate dm-mp relationships 401
The DMA-APM transfer function would ideally be maintained as a constant shape while 402
scanning the log(dm)- log(mp) space in order to minimize skewness induced by the instrument on 403
measurements (Lall et al. 2009). In most of the DMA-APM operations, one operating parameter 404
of either the DMA or the APM (e.g., VDMA, VAPM, or ω) is scanned to measure the particle 405
population in the log(dm)- log(mp) space. This is done in order to obtain the values or 406
distributions of ρeff and Df (McMurry et al. 2002; Park et al. 2003; Malloy et al. 2009; 407
Zangmeister et al. 2014). An example of DMA voltage scanning is the APM-SMPS 408
measurement (Malloy et al. 2009). The shape of the DMA-APM transfer function cannot be 409
maintained in this case because (1) the DMA transfer function continuously changes in the 410
log(dm) space due to Cc (dp) and (2) λc also changes with dm (equation 11). The inversion of the 411
DMA-APM data, which incorporates the DMA-APM transfer function, would be required to 412
resolve this issue. 413
In many applications of the DMA-APM system, an operating parameter of the APM is 414
scanned to measure the particle population of log(dm)- log(mp) while the DMA operating 415
condition is fixed (McMurry et al. 2002; Radney et al. 2013; Zangmeister et al. 2014). An 416
advantage of this scanning method is that the DMA transfer function is maintained throughout 417
the operation, which allows us to focus on the APM transfer function. The resolution and the 418
shape of the APM transfer function should be kept constant during scanning, which is satisfied 419
by keeping λc constant (table 11 and equations 15 and 17). Equations 15 and 17 suggest that the 420 Field Code Changed
Formatted: Font: Not Bold
19
resolution stays the same in the logarithmic scale as long as λc for a certain diameter is kept 421
constant. λc is determined by several parameters, including QAPM, the dimensions of the APM, Zp 422
(dp), and mcω2 (equation 11). QAPM is not scanned for most of the APM operations, and the 423
dimensions of the APM, such as LAPM, cannot be changed during operation. In the case of the 424
DMA-APM system, Zp (dp) can also be considered a constant because the particles are already 425
prescribed by the DMA. 426
A constant λc value can be achieved by keeping mω2 constant. Particle classification by 427
the APM is controlled by both VAPM and ω (equation 10), meaning that mc can be scanned by 428
changing one of them. Equation 10 demonstrates that mcω2 is preserved as long as VAPM and the 429
physical dimensions of the APM are maintained, meaning that λc does not vary when ω is 430
changed to scan mc for a fixed value of VAPM. 431
Figure 88 compares the DMA-APM transfer functions for the ω scan (fixed VAPM) and 432
the VAPM scan (fixed ω). The shape of the DMA-APM transfer function does not change during 433
the ω scan in the log (dp)- log (mp) space. On the other hand, the DMA-APM transfer function is 434
narrower for higher values of mc when VAPM is scanned because λc is proportional to mc (equation 435
11). In conclusion, the ω scan has the advantage of maintaining the DMA-APM resolution 436
compared with the VAPM scan. 437
A caveat for the above discussion is that λc depends on Zp(dm), even though the range of 438
Zp(dm) is narrow following particle classification by the DMA (figure 77). For this reason, the 439
DMA-APM transfer function does not have a rectangular shape in the log(dm)- log(mp) space. 440
This dm dependence in the DMA-APM transfer function needs to be carefully considered when 441
20
interpreting data, especially when a particle population has a uniform mass-mobility relationship 442
(i.e., figures 22a and 22b). When mc is close to 3
m in
1
6e ff
d or m in
fD
fd , the dm of the particles 443
classified by the DMA-APM system is close to dmin. On the other hand, the dm of the classified 444
particles is close to dmax when mc is around 3
m a x
1
6e ff
d or m ax
fD
fd . Even if VAPM is fixed when 445
scanning mc to maintain λc for a certain diameter, the λc corresponding to the classified particles 446
by the DMA-APM system could change due to the fact that the dm of the classified particles 447
depends both on the DMA and the APM. For this reason, the distribution of mp or ρeff , as 448
measured by the DMA-APM system, may not be symmetric, even if VAPM is fixed as a constant. 449
Ideally, the VAPM should be slightly adjusted when scanning mc so that λc is maintained at a 450
certain constant value for the classified particles. However, such an operation requires prior 451
knowledge regarding the mass-mobility exponent. 452
453
4.2. Operating the DMA-APM system to remove multiply charge particles 454
The DMA-APM system is used in some applications as a tool to eliminate multiply 455
charged particles (Pagels et al. 2009; Shiraiwa et al. 2010). In these cases, the DMA-APM 456
transfer function should have a high particle transmittance in region 2C in order to effectively 457
classify the particles of interest. The DMA-APM transfer function, on other other hand, must be 458
sufficiently narrow in order to remove multiply charged particles. However, these two conditions 459
contradict each other. The maximum value of the DMA-APM transfer function is higher for 460
smaller values of λc (Table 33), while the resolution of the DMA-APM transfer function is 461
21
narrower for higher values of λc (section 3.33.3). A method to obtain the maximum value of λc 462
that satisfies these conditions is discussed in the following section, assuming that particle 463
population has a narrow distribution of mass-mobility relationship (figures 22a and 22b). 464
When operating the DMA-APM system to remove multiply charged particles, the 465
central part of the DMA-APM transfer function, which is located at {dp, mp} ={dc,+1, mc, +1}, is 466
adjusted to classify the desired particles. The ρeff corresponding to this point is denoted as ρeff_c,+1. 467
The maximum value of ρeff for multiply charged particles, which is located at P1 for +2 charge 468
particles (P1, +2), must be smaller than ρeff_c,+1 to completely remove the multiply charged 469
particles (Figure 99). These conditions lead to the following equation 470
, 1 _ m a x , 2
3
_ , 1
m in , 2
m in , 2
c o th / 2 1 2
c c e ff
c A P M c
c
d
r dd
d
(22). 471
This equation determines the minimum value of λc to eliminate multiply charged particles, since 472
coth (λc /2) monotonically decreases for higher values of λc. 473
Figure 99 shows an example of a condition that satisfies equation 22. The uniform flow 474
model was used for figure 99a, while the parabolic flow model was employed to calculate the 475
APM transfer function in figure 99b. In both figures, , 1c cd
is equal to 930 kg m-3, and dc,+1 is 476
set at 100 nm. In this case, the doubly charged particles with 930 kg m-3 of ρeff cannot pass 477
through the DMA-APM system because the classification region for +2 particles (dc,+2 = 150.9 478
nm) does not overlap with the area for particle population, which is on the line of of ρeff = 930 kg 479
m-3. 480
22
This condition can be further generalized to non-spherical fractal particles. In that case, 499
1 , 2m
of the APM at dmin,+2 must be smaller than the mass of particles of interest, which equals 500
m in , 2
fD
fd
(equation 3) 501
1, 2 m in , 2 m in , 2
_ , 1
m in , 2
m in , 2
c o th / 2 1 2
f
f
D
f
D
c A P M c
c
m d d
r dd
d
(23) 502
where 2 , 2
m
is assumed to be smaller than
m in , 2
fD
fd
in deriving this equation. Since
1 , 2m
is 503
always larger than 2 , 2
m
(equation 13), no solution is available for equation 23 when this 504
assumption is invalid. Equation 23 is more general than equation 22 because these two equations 505
are equivalent for spherical particles (Df = 3). This equation will be useful for experiments where 506
generation of monodisperse fractal particles is needed, such as a study on the optical properties 507
of soot particles. 508
Interestingly, λc for single and multiple charged particles are the same for the 509
DMA-APM system (equation 11) because their Zp values are the same as long as they are 510
classified by the same DMA (equations 4 and 5). Similarly, mc/qe does not depend on the 511
particle charge (equation 10). An implication of this interesting fact is that the minimum value of 512
λc does not depend on mp or ρeff, as long as dc and dmin are the same. 513
This phenomenon is also useful in considering the elimination of highly (q > 2) charged 514
particles. As evident in figure 9, the condition to eliminate multiply charged particles requires the 515
slope of a line connecting {dp, mp} ={{dc,+1, mc, +1},{dc,+2, mc, +2}} to be smaller than Df in the log 516
Field Code Changed
23
(dp)- log (mp) space. The distance between {dc,+n, mc, +n} (n ≥ 3) and the line for particle 517
population is further than that for doubly charged particles, while λc does not depend on the 518
particle charge (figure S1). The implication is that highly charged particles (q ≥ 3) are always 519
removed by the DMA-APM system when it is being used to eliminate doubly charged particles 520
from the system. 521
522
5. Conclusions 523
The transfer function of the DMA-APM system was developed by overlapping that of 524
the DMA and the APM, and mapped on the log(mp)- log(dp) space. The APM transfer function 525
was calculated using either the uniform or parabolic flow models. The uniform flow model has 526
an analytical expression that is favorable for investigating the instrumental response theoretically. 527
On the other hand, the parabolic flow model provides the APM transfer function more accurately. 528
The mp and ρeff resolutions of the DMA-APM system were theoretically investigated using the 529
derived transfer function. The resolution of the DMA-APM system was also evaluated 530
theoretically. 531
The DMA-APM system is frequently used to measure the ρeff distribution of particles 532
and is occasionally used to eliminate multiply charged particles. The ideal operations of the 533
DMA-APM system for these applications were also discussed. In measuring the mp or ρeff 534
distributions, the system would provide accurate data when the rotation speed of the APM is 535
scanned to measure the distributions because the APM resolution parameter λc does not vary in 536
24
that case. In eliminating multiply charged particles, the minimum value of λc for that application 537
can be calculated using a derived equation. 538
539
Acknowledgement 540
This research was supported by the National Research Foundation Singapore under its Singapore 541
NRF Fellowship scheme (National Research Fellow Award, NRF2012NRF-NRFF001-031), the 542
Earth Observatory of Singapore (EOS), and Nanyang Technological University. I acknowledge 543
useful comments and suggestions from the editor and anonymous reviewers. 544
545
25
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618
619
620
27
Table 1 621
The APM transfer function for the uniform flow model (Ehara et al. 1996). ρ(s) is defined as 622
_2
2 _ 1 _
1
ln /
A P M
c A P M
A P M A P M
Vs r
s r r
. 623
Range APM transfer function Ω
(the uniform flow model)
Region A 1
s s
0
Region B 1 2
s s s 1 1 ex p / 2
cs s
Region C 2 2
s s s e x p
c
Region D 2 1
s s s 1 1 ex p / 2
cs s
Region E 1
s s 0
624
625
626
28
Table 2 627
Dimensions of the APM used for calculations in this study. These values are taken from the design values of APM-3600 628
(KANOMAX Japan, Inc.) 629
Parameter Size (m)
r1_APM 0.05
r2_APM 0.052
LAPM 0.25
630
631
632
29
Table 3 633
The DMA-APM transfer function for the uniform flow model. ρ(mp) is defined as 634
_2
2 _ 1 _
1
ln /
A P M
p c A P M
p A P M A P M
q e Vm r
m r r
. 635
DMA-APM transfer function ,p pm d (the uniform flow model)
Region 1 0 ( no particle passes through the DMA)
Region 2A 0 ( no particle passes through the APM)
Region 2B
, ,
, , , , , ,
1 1 e x p1, 1 , 1 2 , 1
2 2
p q p q c
p p pp q c q p q c q p q c q
m m
Z d d Z d d Z d d
Region 2C , , , , , ,
1, 1 , 1 2 , 1 ex p
2p p p
p q c q p q c q p q c q cZ d d Z d d Z d d
Region 2D
, ,
, , , , , ,
1 1 e x p1, 1 , 1 2 , 1
2 2
p q p q c
p p pp q c q p q c q p q c q
m m
Z d d Z d d Z d d
Region 2E 0 ( no particle passes through the APM)
Region 3 0 ( no particle passes through the DMA)
636
637
30
Figure captions 638
Figure 1. The log(mp)- log(dm) relationships for the particles. (a) ρeff ; (b) Df. 639
Figure 2. Examples of areas for particle population in the log(mp)- log(dm) space. (a) Spherical 640
(or nearly spherical) particles with a constant value of ρeff. ρeff was assumed to be 1000 kg m-3. 641
Particles can only populate on the black solid line; (b) Aggregated non-spherical particles with a 642
certain value of Df (e.g., soot). The black solid line on which particles can populate was 643
calculated as 6 2 .4 1
6 1 0p m
m d
based on Park et al. (2003); (c) A mixture of spherical and 644
non-spherical particles with a range of ρeff. Particles may populate in the shaded area. 645
Figure 3. Examples of (a) the DMA and (b) the APM transfer functions. The DMA transfer 646
function was calculated at dc = 100 nm for β = 0.1. The following parameter set was used to 647
calculate the APM transfer function: VAPM = 100 V, ω = 523.599 rad s-1 (equivalent as 5000 rpm), 648
QAPM = 1.67×10-5 m3 s-1 (equivalent as 1 l min-1), q=1, and dm = 100 nm. 649
Figure 4. Diameter dependences of (a)c
m , 1
m , and
2m
, and (b)λc. The following parameter set 650
was employed to obtain these values: VAPM = 100 V, ω = 523.599 rad s-1, QAPM = 1.67×10-5 m3 651
s-1, and q=1. 652
Figure 5. Seven different regions for the DMA-APM transfer function. 653
Figure 6. Examples of the DMA-APM transfer functions calculated using (a) the uniform flow 654
model and (b) the parabolic flow model. The following parameter set was employed for the 655
calculations: VAPM = 100 V, ω = 523.599 rad s-1, QAPM = 1.67×10-5 m3 s-1, q=1, dc = 100 nm and 656
β = 0.1. 657
31
Figure 7. Comparison of diameter resolutions of the DMA and APM for particles with a uniform 658
value of ρeff. Grey dash lines show important values for the transfer functions of the DMA and 659
the APM, including c
m , 1
m , dc, dmin, and dmax. A black dashed line for ρeff, which corresponds 660
to the value in the central part of the classification region (ρeff = 930 kg m-3), is also shown. If all 661
the particles populate on the line of ρeff = 930 kg m-3, then particles with 1 1
pm m m
can be 662
classified by the system. The corresponding diameter range (m in m axA P M m A P M
d d d ) is narrower 663
than the particle classification range by the DMA (m in m axm
d d d ). The colored area represents 664
the DMA-APM transfer function, which is calculated at VAPM = 85 V, ω = 523.599 rad s-1, QAPM 665
= 5.0×10-6 m3 s-1, q=1, dc = 100 nm and β = 0.1. See the text for further details. 666
Figure 8. Comparisons of the APM scanning methods. (a~c) shows the DMA-APM transfer 667
functions for rotation speed scanning and (d~f) corresponds to voltage scanning. These transfer 668
functions were calculated for QAPM = 1.67×10-5 m3 s-1, q=1, dc = 100 nm and β = 0.1. The 669
parameter sets of {VAPM, ω} are (a) {100 V, 641.274 rad s-1}, (b) {100 V, 523.599 rad s-1}, (c) 670
{100 V, 427.516 rad s-1}, (d) {66.667 V, 641.274 rad s-1}, (e) {100V, 641.274 rad s-1}, and (f) 671
{150 V, 641.274 rad s-1}. 672
Figure 9. Elimination of multiply charged particles by the DMA-APM system. (a) the uniform 673
and (b) the parabolic flow models were used for the calculation. The following parameter set was 674
employed for the calculations: VAPM = 85 V, ω = 523.599 rad s-1, QAPM = 3.33×10-5 m3 675
s-1(equivalent as 2 l min-1), dc,+1 = 100 nm and β = 0.1. The DMA-APM transfer function for +2 676
particles does not overlap with the line for ρc of +1 particle (930 kg m-3). 677