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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].

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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

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