Design Considerations and Performance Evaluation of a Compact Aerosol Particle Mass Analyzer

Abstract

A compact aerosol particle mass analyzer (APM) of which the size of the classifier was significantly reduced than that of the first commercial model (Kanomax Model 3600) was developed. Firstly, requirements for desired performance in classifying particle mass were set forth. Secondly, a theoretical framework for the design parameters of an APM that satisfies the requirements was formulated. Thirdly, the design parameters were determined that satisfies the requirements while reducing the instrument size. The requirements include the condition that the classification range covers from 0.001 to 1000 fg (approximately 12 to 1200 nm in size for spherical particles having the density of 1 g/cm³), and the condition that both the classification resolution and particle penetration in this mass range are higher than certain specified values. A prototype having the design parameters determined according to this theoretical framework was constructed, and its performance was evaluated experimentally. The external dimensions of the electrodes of the compact APM are approximately 140 mm in length and 60 mm in diameter. It was confirmed that the performance of the compact APM operated at the aerosol flow rate of 0.3 L/min was comparable to that of the Model 3600 APM operated at 1 L/min. Because of the reduced size and of the resultant improved portability, it is expected that the compact APM is readily applicable to field measurements.

Copyright 2013 American Association for Aerosol Research

1. INTRODUCTION

The aerosol particle mass analyzer (APM) classifies aerosol particles according to their mass-to-charge ratio (Ehara et al. 1996). When the number of charges on a particle is known, the APM can determine the particle mass. The APM has been used, often in combination with a differential mobility analyzer (DMA), for evaluation of various particle properties such as “effective” density (McMurry et al. 2002; Geller et al. 2006; Malloy et al. 2009; Liu et al. 2012), morphology through estimation of the mass-mobility exponent (Park et al. 2003a; Ku et al. 2006; Kim et al. 2009; Nakao et al. 2011; Scheckman and McMurry 2011; Torvela et al. 2011; Eggersdorfer et al. 2012; Guha et al. 2012; Ku and Evans 2012), the mixing ratio of internally-mixed particles (Sakurai et al. 2003; Park et al. 2008; Khalizov et al. 2009; Ma and Zachariah 2009), material density (Park et al. 2004; Kuwata et al. 2012), particulate mass concentration (Park et al. 2003b; Saito et al. 2008), mass (Kuwata et al. 2009; Lall et al. 2009), volume (Lall et al. 2008), and porosity (Lee et al. 2011). The APM can also be used as a generator of particles having a specified mass (Malik et al. 2011; Moteki et al. 2011; Laborde et al. 2012; Beranek et al. 2012). A variant of the APM called the Couette Particle Mass Analyzer (CPMA) has been developed (Olfert and Collings 2005; Olfert et al. 2006), and used for similar purposes (Olfert et al. 2007; Cross et al. 2010).

In our previous article (Tajima et al. 2011), we reported the appropriate mass range and desired operating conditions for the first commercial instrument, Model 3600 APM (Kanomax Japan Inc.), from the viewpoint of maximizing the mass classification performance. This model of APM is, however, significantly large in size, and hence it is not very suitable for field measurements. In the present study, we have developed an APM that is significantly compact and more suitable for use in field measurements compared to the Model 3600 APM.

In Section 2, starting with laying out a set of requirements for mass classification performance, we derive constraint formulas for the design parameters of the APM. We then determine a combination of the design parameters realizing an APM which is significantly smaller in size than and yet comparable in performance to the Model 3600 APM. In Section 3, we present experimentally evaluated performance of a prototype APM that was constructed according to the determined set of design parameters. The performance in terms of the accuracy and resolution in classifying particle mass and of the particle penetration through the APM is compared with that of the Model 3600 APM.

2. THEORETICAL CONSIDERATIONS

2.1. Design and Operation Parameters

The APM operates by balancing the electrostatic and centrifugal forces acting on charged particles introduced in rotating cylindrical electrodes. In the theoretical analysis that follows, we assume for simplicity that particles carry a single elementary charge. A brief discussion about the effect of multiply charged particles on mass distribution spectra obtained with the APM is given in the online supplemental information. Under the assumption of single charge, we may consider that the APM classifies particles according to the mass.

The performance of mass classification by the APM is characterized by the transfer function, which is defined as the ratio of the number flux of particles with a specified mass m at the outlet of the APM to that at the inlet when the APM is operated at a constant voltage V and angular velocity ω of the electrodes. The APM transfer function derived theoretically under the assumption that the velocity profile of the aerosol flow through the electrode gap is either uniform or parabolic is shown in Figure 1. The center of the mass band of the transfer function m _c (to be more strict, the particle mass at which the centrifugal force acting on a particle at the radial distance r_c balances with the electrostatic force) is given by

where e is the elementary charge, r ₁ and r ₂ are the inner and outer radius of the electrode gap, and r _c = (r ₁ + r ₂)/2. In order to the classification resolution to be reasonably high, it is required that δ/r _c ≪ 1, where δ = (r ₂ – r ₁)/2 (Equation (5)). We assume in the following analyses that this condition is satisfied. Equation (1) is then approximated as

FIG. 1 Theoretical APM transfer functions when the velocity profile of the aerosol flow is assumed uniform (solid line) and parabolic (dashed line). The transfer functions for λ_c = 0.47 are shown in this figure. (Reproduced from Weingartner et al. (2011) with permission of John Wiley & Sons, Inc.)

The trajectories of particles having a mass m and a mechanical mobility B through the electrode gap, and hence the shape of the transfer function, are determined solely by the non-dimensional quantity termed the classification performance parameter

where L is the axial length of the electrode gap, and is the mean velocity of the aerosol flow through the electrode gap. The latter is given approximately by with Q being the aerosol volumetric flow rate. It should be noted that λ depends on the particle properties m and B. We are interested only in particles whose mass is located within the mass band of the transfer function. When the width of this band is sufficiently smaller than m _c, which we assume in the following, we can replace m in Equation (3) with m _c. In addition, we replace the mobility B of particles whose mass is in this band with a constant value B_c . This approximation is valid either when the particle density is uniform and there is one-to-one correspondence between m and B, or when a DMA is installed upstream of the APM and the mobility is precisely specified. In the present study, we assume the former case; to be more specific, we assume that particles are spheres having the density of 1.0 g/cm³. Under these assumptions, λ can be approximated by

where B(m _c) denotes the mechanical mobility corresponding to m _c. Note that, unlike λ, the value of λ_c is determined uniquely once the operating conditions are specified, and that it does not depend on particle properties.

Because the uniform flow model for the aerosol velocity profile allows us an analytical expression for the theoretical transfer function, we adopt this model in the following analysis. Later in Section 2.5, we will confirm that the result obtained in this model is also valid for the parabolic flow model. Let the resolution, R, be defined as the ratio of m _c to the full base length of the transfer function band, Δm. As Figure 1 indicates, R is given by

Let us define the particle penetration, T, as the maximum height of the transfer function, which is given by

In general, we consider that the larger the values of R and T are, the better the APM performance is.

The parameters that determine m _c, R, and T can be categorized into the following two groups ¹ .

Operation parameters: ω and V

Design parameters: r _c, δ, L, and Q

The parameters in the first group are those which can be selected at each operation from within certain limits; the limits are formally represented by

We chose the values of these limits as

The value of V _hi was chosen to avoid dielectric breakdown between the APM electrodes, and the value of V _lo was set roughly at 10 times the resolution of a readily available digital-to-analog converter to be used for controlling the electrode voltage. The value of ω _hi was roughly the maximum capacity of a relatively compact motor typically used to rotate the electrodes. Finally, the value of ω _lo was determined from the requirement that the relative precision of the motor in controlling the rotational velocity of the electrodes be better than 3%, as the rotational velocity fluctuates unacceptably when the electrode is rotated even slower.

Determining the second group parameters is the target of the present study. The two parameters, L and Q, can be reduced to a single parameter l = L/Q, because L and Q appear only in the combined form L/Q in λ_c.

2.2. Performance Requirements

We set forth the following formal requirements that need to be satisfied by a compact APM.

Requirement A: The variable range of m _c covers entirely a specified range (m _lo, m _hi).

Requirement B: For any m _c in (m _lo, m _hi), there exists an operating condition under which R is larger than or equal to a specified value R _lo.

Requirement C: For any m _c in (m _lo, m _hi), there exists an operating condition under which T is larger than or equal to a specified value T _lo.

Here, it is understood in Requirements B and C that there must be an operating condition such that these two requirements are simultaneously satisfied. The actual values of m _lo, m _hi, R _lo, and T _lo are chosen so that the performance of the compact APM is comparable to that of the Model 3600 APM; namely,

The mass interval in Equation (11) corresponds approximately to the size interval of (12 nm, 1.2 μm) for spherical particles having the density of 1.0 g/cm³.

Because the three requirements above do not result in a unique combination of design parameter values, it is desirable to introduce additionally a performance measure that can further narrow the range of the desired parameter values. Such a measure could be the product RT, or a product of certain powers of R and T. It would be, however, significantly complicated, though not impossible, to optimize such a quantity while satisfying the Requirements A to C, because these quantities depend not only on the design parameters but also on the operation parameters. Instead, we simply use δ/r _c as such an additional performance measure. Equations (5) and (6) indicate that if there are multiple combinations of the design parameters that realize the same value of T, a smaller value of δ/r _c will lead to a larger value of R, which rationalizes the use of δ/r _c.

2.3. Constraints on Design Parameters

In this section, a theoretical framework is developed to derive possible combinations of the design parameters that satisfy the above three requirements. First we note that, for fixed design parameters, the values of m _c and λ_c are fully determined from ω and V. Figure 2 shows typical contours of m _c and of λ_c in the (ω, V) plane. The rectangle in Figure 2 shows the operation region specified by Inequalities (7) and (8). We can select any two parameters out of the four parameters, m _c, λ_c, ω, and V, as independent variables, and regard the remaining parameters as functions of the independent variables. Here, we choose (m _c, λ_c) as independent variables. The operation region in the (m _c, λ_c) plane appears a distorted parallelogram as shown in Figure 3.

FIG. 2 Typical contours of m _c (solid curves) and λ_c (dashed curves) in the (ω, V) plane. The rectangle ABCD indicates the operation region.

FIG. 3 Schematic illustration of the operation region in the (m _c, λ_c) plane with both axes represented in logarithmic scale. The distorted parallelogram ABCD corresponds to the rectangle ABCD in Figure 2.

Let Min(m _c) and Max(m _c) denote the minimum and maximum values of m _c in the operation region. Requirement A can be written as

with the understanding that these two conditions must be simultaneously satisfied. As is obvious from Figure 3, Min (m _c) occurs at point A where (ω, V) = (ω _hi, V _lo); namely, Min (m _c) = m _c(A). Since m _c(A) = eV _lo/(2r _cδω² _hi) from Equation (2), In equality. (13) is rewritten as

Similarly, since Max(m _c) =m _c(C) = eV _hi/(2r _cδω² _lo); Inequality (14) is rewritten as

The symbols [RA1] and [RA2] are to indicate that these are conditions derived from Requirement A.

In order to the Requirements B and C to be satisfied, there must exist in the operation region some value of λ_c such that the inequalities

simultaneously hold true for any m _c in (m _lo, m _hi). Here, the symbols R(λ_c) and T(λ_c) indicate explicitly that R and T are functions of λ_c, as expressed by Equations (5) and (6). Since R(λ_c) and T(λ_c) are monotonically increasing and decreasing functions of λ_c, respectively, Inequalities (15) and (16) can be replaced with

where

As seen in Figure 3, for a given m _c, λ_c spans across a range from Min (λ_c|m _c) to Max (λ_c|m _c) under the constraints of (7) and (8), such as from points F to E in the figure. Here, Max (|m _c) and Min (|m _c) indicate, respectively, the conditional maximum and minimum for a given m _c. For a given m _c, if the inequality

is met, then there exists λ_c that satisfies Inequality (15′). Similarly, if

is met, then there exists λ_c that satisfies Inequality (16′). If

holds true in addition to Inequalities (15^″) and (16^″), there exists an operating condition such that Inequalities (15′) and (16′) are simultaneously satisfied. In other words, the three inequalities, (15^″), (16^″) and (18), are necessary and sufficient conditions for Requirements B and C to be satisfied simultaneously.

Now we rewrite the conditions (15^″) and (16^″) into inequalities in terms of the design parameters. For Inequality (15^″), as Figure 3 shows, if Inequality (15^″) holds both at m _c = m _hi and at m _c = m _lo, it automatically holds at any m _c within (m _lo, m _hi). Hence, we have

Since Max (λ_c|m _lo) occurs at point E in Figure 3, we have Max (λ_c|m _lo) = λ_c(E) = 8πm _lo B(m _lo)ω _hi ² r _cδl, while λ_lo can be approximated as from our assumption δ/r _c ≪ 1. Hence Inequality (15^″-1) is rewritten as

Similarly, noting that Max(λ_c|m _hi) = λ_c(H) = 4πeV _hi B(m _hi)l, we can rewrite Inequality (15^″-2) as

The symbols [RB1] and [RB2] indicate that these are conditions derived from Requirement B. In the same manner, Inequality (16^″) can be rewritten into

Finally, Inequality (18) can be rewritten from Equation (17) as

The symbol [RBC] indicates that this condition is related to the simultaneousness of Requirements B and C.

In summary, the design parameters must satisfy [RA1], [RA2], [RB1], [RB2], [RC1], [RC2], and [RBC], in order for Requirements A, B, and C to be fulfilled.

2.4. Optimum Design

Substituting the limit values given in Equations (9)–(12) into the conditions [RA1] to [RBC], we have

	[RB1]    r 2 c l ≥ 8.390 s,
	[RB2]    r c l/δ ≥ 0.1894 s/mm2,
	[RC1]    l ≤ 0.4320 s/mm2,
	[RC2]    r c δ l ≤ 6.371 s,
	[RBC]   δ/r c ≤ 0.4091.

The condition [RBC] need not be considered, because it is automatically satisfied under the assumption δ/r _c ≪ 1. We additionally assume here that practically appropriate values of L and Q are, respectively, in the ranges of 40–200 mm and of 0.3–1.0 L/min. The corresponding range of l is then 0.003–0.040 s/mm². The condition [RC1] is automatically satisfied, if l is in this range. In addition, the condition [RC2] is automatically satisfied, if [RA2] is satisfied and l is in this range. From these arguments, the conditions we have to consider are now reduced to [RA1], [RA2], [RB1], and [RB2]. It may be worth pointing out here that the conditions [RA1] and [RA2] were derived from the requirement on m _c, while the conditions [RB1] and [RB2] were from the requirement on R.

The regions in the (r _c, δ) plane that satisfy the conditions [RA1] to [RB2] for fixed values of l are shown in Figure 4a–d. As l increases, the lower boundary of r _c decreases. If we adopt Q = 0.3 L/min which is an aerosol flow rate often employed for relatively small flow rate condensation particle counters (CPCs) and optical particle counters (OPCs), and if we choose L = 100 mm as a reasonably short electrode length, we have l = 0.02 s/mm². Considering that a small r _c is desirable in realizing a compact APM, and that a small value of our performance measure, δ/r _c, is desirable, we chose the final combination of the design parameter values as l = 0.02 s/mm² (L = 100 mm, Q = 0.3 L/min), r _c = 24.5 mm, and δ = 0.5 mm. With this value of δ, the maximum electric field strength that may occur in the electrode gap, V _hi/(2δ), is 2×10⁶ V/m, which is smaller than the commonly accepted threshold value of the dielectric breakdown of air, 3×10⁶ V/m (Lide [CRC Handbook of Chemistry and Physics], 2008). The chosen combination of the design parameter values is marked by a cross in Figure 4c. In this choice, the value of the performance measure, δ/r _c, is 0.0204, which is comparable to the value, 0.0196, for the value for the Model 3600 APM. Note that this value of δ/r _c is consistent with δ/r _c ≪ 1 assumed in the above analysis.

FIG. 4 Regions in the (r _c, δ) plane satisfying the conditions [RA1], [RA2], [RB1], and [RB2] for (a) l = 0.003 s/mm², (b) 0.01 s/mm², (c) 0.02 s/mm², and (d) 0.04 s/mm². The hatched areas indicate the regions in which all four conditions are satisfied. The crossmark in panel (c) represents the combination of the design parameters finally selected. Note that [RA1] and [RA2] represent the requirements, derived from Inequalities (13) and (14) respectively, on the minimum and maximum particle masses that can be classified by the APM, while [RB1] and [RB2] the requirements, both derived from Inequality (15), that the resolution at m _c = m _lo and m _c = m _hi, respectively, be larger than R _lo (= 0.5).

We note from Figure 4c that the conditions [RA2] and [RB2] are satisfied by wide margins. Taking advantage of this fact, we reselect the value of ω _lo at 1000 rpm, instead of 500 rpm as initially assumed. The operation region of the APM with this value of ω _lo is shown in Figure 5. A summary of comparison between the APM thus designed and the Model 3600 APM is given in Table 1.

Table 1 Comparison between the compact APM developed in the present study and the Model 3600 APM

Electrode Dimensions		Compact APM	Original APM MODEL 3600
Radius, r c	(mm)	24.5	51
Gap, 2δ	(mm)	1	2
Length, L	(mm)	100	250
Rotational velocity	(rpm)	1000–14,000	50–9500
Electrode voltage	(V)	0.3–2000	0.3–2000
Aerosol flow rate (recommended value)	(L/min)	0.3	0.3–1.5
Particle mass range^a	(fg)	9.1 × 10^−4–1.2 × 10^3	4.8 × 10^−4–5.9 × 10^4
Particle size range^b	(nm)	12–1300	9.7–4800
^aTheoretical range satisfying R (resolution) > 0.5 and T (penetration) > 0.1, simultaneously.
^bDerived from the mass range assuming that the density is 1 g/cm^3.
^cRange calculated for the aerosol flow rate of 1.0 L/min.

FIG. 5 Operation region of the compact APM constructed in the present study (distorted parallelogram). The hatched area indicates the region in which Requirements A, B, and C are all satisfied. The symbols ○, □, and ▪ represent the points on which APM spectra were experimentally obtained, where ○ represents an experiment conducted in configuration (a) with PSL particles used as test particles; □ in configuration (b) with PSL particles; and ▪ in configuration (b) with Santovac oil particles. See Figure 6 for configurations (a) and (b).

2.5. Validity in the Parabolic Flow Model

The preceding analysis is based on the uniform flow model for the velocity distribution of the aerosol flow in the electrode gap. A more realistic model of the flow profile is the parabolic flow model. In this section, we examine whether the design parameters chosen above satisfy Requirements A to C in the parabolic flow model.

First we note that the expressions of m _c and λ_c (Equations (2) and (4)), and hence the location of the operation region in the (m _c, λ_c) plane are model independent. Consequently, Requirement A is satisfied irrespective of the flow model. On the other hand, R(λ_c) and T(λ_c), and hence the values of λ_lo and λ_hi, depend on the flow model. Equation (17) gives λ_lo and λ_hi in the uniform flow model, and with the chosen values of the design parameters, we have

In the parabolic flow model, R(λ_c) and T(λ_c) need to be calculated numerically. By performing this calculation, we obtain

Because the interval (λ_lo, λ_hi) of the parabolic flow model encompasses that of the uniform flow model, it is obvious that Inequalities (15^″), (16^″), and (18), and hence Requirements B and C, are satisfied also in the parabolic flow model.

3. EXPERIMENTAL VERIFICATION

3.1. Method

To confirm the validity of the preceding theoretical treatment, an APM having the values of the design parameters thus determined was constructed, and its performance was experimentally evaluated and compared to that of the Model 3600 APM. In typical measurements using an APM, the particle concentration at the APM outlet, C_N , is recorded for a series of electrode voltage, V, at constant ω. We call a set of C_N –V pairs an APM spectrum ² . In the present study, we compare experimental APM spectra obtained for particles with a known mass to the corresponding theoretically predicted spectra. The experiment was carried out in either of the two configurations illustrated in Figure 6a and b. In configuration (a), the particle concentration at the APM inlet was measured before and after data collection of C_N by switching a three-way valve connected to the inlet of a CPC. In configuration (b), two CPCs were used to measure the concentrations at the inlet and outlet of the APM simultaneously. In either configuration, the electrode voltage was swept linearly with time.

FIG. 6 Two experimental configurations.

We proposed in the previous paper (Tajima et al. 2011) a scheme to select an appropriate value of λ_c for APM operations, which may be termed the “resolution matching scheme.” This scheme assumes the use of a DMA in combination with the APM, and the value of λ_c is so selected that the resolution of the APM expressed in terms of particle mobility matches the resolution of the DMA. In the present study, the ratio of the aerosol to sheath air flow rates of the DMA was set at 0.1, and the corresponding λ_c calculated according to the resolution matching scheme ranged from 0.25 to 0.5 depending on the particle size. In the experiment, we adopted λ_c = 0.23 and 0.51, the values roughly equal to the limits of this range, as long as they were inside the operation region (Figure 5).

Polystyrene latex (PSL) particles of eight sizes; 29, 48, 70, 76, 100, 309, 506, and 791 nm (STADEX; JSR Co.), and Santovac oil particles were aerosolized to be used for test particles. When PSL particles were used, the DMA voltage was adjusted so that the highest particle concentration was attained at the DMA outlet. The density of each PSL particle type is given by the manufacturer, and is in the range from 1.054 to 1.115 g/cm³. Santovac oil particles were aerosolized by an electrospray aerosol generator (Model 3480, TSI Inc.), and were selected by the DMA at five different sizes ranging from 12 to 35 nm. We assume that the density of the Santovac particles is identical to that of a bulk liquid, 1.198 g/cm³. The mass of the test particles ranges from 0.0011 to 274 fg, which covers a major part of the interval specified by Equation (11).

The symbols in Figure 5 (○, □, and ▪) indicate the points on which APM spectra were collected in the experiment. Since the line λ_c = 0.23 for m _c smaller than about 10⁻² fg is outside the operation region, the operation conditions for particles with such small m _c were chosen so that ω took its maximum possible value, ω _hi (= 14,000 rpm).

Prior to the experiment, continuous operations at 14,000 rpm longer than 10 h were conducted several times, and it was confirmed that the constructed APM worked stably at this maximum rotational speed. Also, it was confirmed that at the maximum voltage (2000 V), the electrostatic breakdown did not occur during the operations.

3.2. Results

The APM spectra obtained experimentally are shown in Figures 7a (PSL particles with λ_c = 0.51), b (PSL particles with λ_c = 0.23), and c (Santovac oil particles). Theoretically, an APM spectrum can be calculated according to

where n(m) = dC_N /dm is the mass distribution of particles at the APM inlet, and Ω_APM(m; V) is a function of m with V as a parameter, representing the APM transfer function for singly-charged particles when the classification voltage is V. In the present study, we neglect the effects of particle Brownian motion on the transfer function (Hagwood et al. 1995). Assuming n(m) to be described by a certain function containing a few adjustable parameters, and fitting C_N (V) in Equation (19) to an experimental spectrum, we can determine n(m). Here, we adopt an asymmetric normal distribution for n(m). For more details of the procedure of determining n(m), see Tajima et al. (2011). The solid curves in Figure 7 represent C_N (V) after the least square fitting was performed. Almost exact agreement between C_N (V) and the experimental spectra observed in Figure 7 suggests the appropriateness of adopting the asymmetric normal distribution as n(m).

FIG. 7 APM spectra for (a) PSL particles with λ_c = 0.51, (b) PSL particles with λ_c = 0.23, and (c) Santovac oil particles. Experimental spectra are shown with symbols, calculated spectra obtained by the least square fitting method shown with solid curves, and theoretically predicted spectra shown with dotted curves. Associated with each experimental spectrum is the values of the DMA-classified diameter and of the number-averaged mass. For spectra of Santovac oil particles, the values of λ_c are also shown.

The number-averaged mass can be calculated from n(m) thus determined. Figure 8 shows the ratio of the number-averaged mass obtained in this way to that predicted theoretically. The latter was derived from the volume of a sphere having the DMA-classified diameter multiplied by the known particle density. Because no noticeable systematic differences in the experimental results were observed between the two experimental configurations depicted in Figures 6a and b, as well as between the two choices of λ_c, the data in Figure 8 were plotted without discrimination between these experimental conditions. For the purpose of comparison, Figure 8 also shows the data obtained in our previous study using the Model 3600 APM operated at the aerosol flow rate of 1 L/min for PSL and NaCl particles (Tajima et al. 2011).

FIG. 8 Ratio of the number-averaged mass obtained experimentally to that predicted theoretically. The theoretically predicted mass values were based on the assumption that particles, including NaCl particles in the experiment with the Model 3600 APM, are spherical.

Figure 8 shows that the experimentally determined mass is in good agreement with the theoretically predicted mass for particles larger than 50 nm (approximately 0.1 fg in mass); the ratio between them is within the interval of 1 ± 0.1, with a slightly larger scatter among data for the prototype compact APM than for the Model 3600 APM. However, for particles smaller than 30 nm (approximately 0.02 fg in mass), a systematic deviation of the ratio from unity is observed, with the measured mass smaller than the predicted mass. This phenomenon is similarly observed for the Model 3600 APM, although quantitative comparison is not possible because different type of particles, especially in terms of their shape, were used in the present and previous studies. We note that the disagreement between theory and experiment in the relatively smaller mass range is observed regardless of the operating conditions as represented by λ_c, as well as regardless of the APM design.

Particle Brownian motion within the APM is expected to shift the APM transfer function to the larger mass side (Hagwood et al. 1995), and hence it cannot explain the observed disagreement. In our separate study (Sakurai et al. 2010), we investigated size change of Stantovac and poly-alpha-olefin (PAO) particles before and after the APM by using a scanning mobility particle sizer, but observed no size change even at 10 nm. Therefore, evaporation of the oil particles is not considered to be the major cause of the disagreement. ³ Larriba et al. (2011) used a high-resolution DMA and a mass spectrometer in tandem to measure the mobility and mass of clusters of an ionic liquid. They showed that the mobility diameter, calculated from the measured mobility using the Stokes–Millikan law, was in excellent agreement with the sum of the volume diameter, which generalizes the mass diameter for binary substances such as salts, and 0.3 nm that accounts for the finite size of air molecules in the air-particle collision cross section (also see Ku and de la Mora [2009] for the 0.3 nm correction). If we apply this 0.3 nm correction to the mobility diameter of 12 nm which is the smallest particle size in our experiment, the predicted mass would decrease by about 7.3%. This correction can, therefore, partly explain the observed disagreement, but not all of it, which amounts to 30 to 80% at 12 nm.

Although no definitive explanation is obtained yet, the observed disagreement might be ascribable to nonsphericity of the test particles in the relatively small size range, which can lead to an increase in mobility equivalent diameter as compared to the diameter of spherical particles having the same mass. It is recently reported on the basis of observation with a scanning electron microscope that the sphericity of the 30 nm PSL particles is not as good as the 50 nm and larger PSL particles (Hayashida 2013). Also, oil droplets moving through the sheath air of a DMA could be distorted depending on their sizes, although we are unable at present to quantitatively estimate this effect.

Besides the locations of the APM spectra, their shapes, especially the widths of the spectra, are of our concern, because they reflect the actual resolution of the instrument. If a reasonable estimate of the shape of n(m) is available, we can calculate a theoretical spectrum according to Equation (19) and compare it with experiment. Because the difference in number-averaged mass between the theoretical prediction and the experimental observation is significant for particles smaller than 30 nm, as we have just seen above, we temporarily use here an effective density such that the predicted number-averaged mass reproduces the observed number-averaged mass. In addition, we assume that the shape of n(m) is derived from the DMA transfer function multiplied by the size distribution at the DMA inlet; the latter for the PSL particles is approximated by a normal distribution having the size standard deviation provided by the particle manufacturer, and for Santovac particles approximated by a uniform distribution (Tajima et al. 2011). The theoretical spectra thus obtained are shown by the dotted lines in Figure 7.

Figure 7 shows that the shapes of the experimental spectra for particles larger than 100 nm are in excellent agreement, both in their widths and heights, with those of the theoretical spectra. The agreement is still good for particles as small as about 50 nm, but for even smaller particles, especially particles smaller than 20 nm, notable broadening is observed in the experimental spectra. Again, the reason for this broadening is not identified at this moment. It should be pointed out that the apparently sharp spectrum for 48 nm PSL particles in Figure 7a was obtained at the rotational velocity almost equal to the maximum velocity ω _hi (see also Figure 5 for the operation conditions). This indicates that an instability of the aerosol flow through the APM does not occur even at ω _hi, and is not the reason for the broadening observed below 20 nm.

The ratios of the full width at half maximum and the peak height of the experimental spectra to those of the theoretically predicted spectra, plotted in Figures 9a and b, respectively, confirm that for particles larger than 50 nm the spectral broadening is not significant. Figures 9a and b also show that both the widths and heights of the spectra obtained with the compact APM are in slightly better agreement with the theoretical predictions than those with the Model 3600 APM.

FIG. 9 Ratios of (a) the full width at half maximum and (b) the peak height of the experimental spectra to those of the theoretically predicted spectra.

The analyses here demonstrate that the constructed APM operated at the aerosol flow rate of 0.3 L/min has a performance comparable to that of the Model 3600 APM operated at 1.0 L/min.

4. CONCLUSIONS

A theoretical framework was formulated that determines the design parameters of an APM which is significantly smaller than the existing Model 3600 APM and yet satisfies several performance criteria. A prototype compact APM having the design parameters determined according to this formulation was constructed, and its performance was experimentally evaluated.

The number-averaged mass of particles larger than 50 nm (approximately 0.1 fg in mass) determined experimentally was in good agreement with theoretically predicted values. The scatter of the experimentally obtained number-averaged mass around the theoretical prediction was slightly larger than that obtained with the Model 3600 APM. For particles smaller than 30 nm (approximately 0.02 fg), a systematic and significant underestimate of the number-averaged mass as compared with theoretical prediction was observed. This phenomenon was observed also for the Model 3600 APM, and regardless of the operating conditions. Although the exact reason for this phenomenon is not identified yet, a possible role of the nonsphericity of the test particles in the observed phenomenon is suggested.

The shape of the observed spectra was in excellent agreement with theoretical prediction, both in terms of their widths and heights, for particles larger than 100 nm, and in fairly good agreement for particles larger than 50 nm. Agreement of the spectral shape between theory and experiment was somewhat better than that of the Model 3600 APM.

These analyses show that the performance in classifying particle mass of the compact APM operated at the aerosol flow rate of 0.3 L/min is approximately the same as that of the Model 3600 APM operated at 1.0 L/min. With the external dimensions of the electrodes being 140 mm in length and 60 mm in diameter, the new compact model of APM can be adopted easily to field measurements.

Acknowledgments

The authors would like to thank Professor Yoshio Otani of Kanazawa University for his helpful support.

[Supplementary materials are available for this article. Go to the publisher's online edition of Aerosol Science and Technology to view the free supplementary files.]

Notes

There is some arbitrariness in categorizing Q in either group, but it is convenient to regard Q as a design parameter for the purpose of determining the design of an APM. Note that Q can be used to control R and T at each APM operation, but the same role can be played also by ω.

In practice, the ratio of C_N to the particle concentration at the APM inlet is often recorded in order to compensate the variation of the concentration at the inlet during the scan over V. We also call a set of “normalized” C ^″ _N–V pairs an APM spectrum.

However, the aerosol temperature at the APM outlet was observed to increase by about 10 degrees at 14,000 rpm operation. Due attention should be paid to the temperature rise when semivolatile particles are to be measured.