Comparison of three essential sub-micrometer aerosol measurements: Mass, size and shape

Abstract

An instrumental trifecta now exists for aerosol separation and classification by aerodynamic diameter (D_ae), mobility diameter (D_m) and mass (m) utilizing an aerodynamic aerosol classifier (AAC), differential mobility analyzer (DMA) and aerosol particle mass analyzer (APM), respectively. In principle, any combination of two measurements yields the third. These quantities also allow for the derivation of the particle effective density (ρ_eff) and dynamic shape factor (χ). Measured and/or derived deviations between tandem measurements are dependent upon the configuration but are generally <10%. Notably, nonphysical values of χ (<1) and ρ_eff (>bulk) were determined by the AAC-APM. Harmonization of the results requires the use of χ in the determination of m and D_m from the AAC-DMA and AAC-APM requiring either a priori assumptions or determination from another method. Further errors can arise from assuming instead of measuring physical conditions – e.g., temperature and pressure affect the gas viscosity, mean free path and the Cunningham slip correction factor therefore impacting D_m and D_ae – but are expected to have a smaller impact than χ. Utilizing this triplet of instrumentation in combination allows for quantitative determination of χ and the particle density (ρ_p). If the bulk density is known or assumed, then the packing density can be determined. The χ and ρ_p were determined to be 1.10 ± 0.03 and (1.00 ± 0.02) g cm⁻³, respectively, for a water stabilized black carbon mimic that resembles aged (collapsed) soot in the atmosphere. Assuming ρ_bulk = 1.8 g cm⁻³, a packing density of 0.55 ± 0.02 is obtained.

Copyright © 2020 American Association for Aerosol Research

EDITOR:

• Jason Olfert

1. Introduction

Physical and morphological properties (size, effective density and shape factor) and the number density of particles are required to predict aerosol behavior. Measuring these parameters accurately at the sub-micrometer level poses many challenges and several methods currently exist to quantify these properties. Most investigations use a differential mobility analyzer (DMA) – first demonstrated in 1975 (Knutson and Whitby 1975) to classify particles based upon their mobility diameter (D_m); i.e., the measured diameter that has the equivalent mobility of a spherical particle with a single net charge (q = ±1) in an electric field. The invention and development of the aerosol particle mass analyzer (APM) in 1996 (Ehara, Hagwood and Coakley 1996) and the centrifugal particle mass analyzer (CPMA) in 2005 (Olfert 2005) has allowed for the additional classification of particles by mass (m) by the balance of centrifugal and electrostatic forces. The combination of D_m and m measured by a DMA in tandem with an APM or CPMA allow for the calculation of effective density (ρ_eff), mass-mobility exponent (D_fm) and effective dynamic shape factor (χ_eff) for particles <1 μm (Ehara et al. 1996; McMurry et al. 2002; Olfert, Symonds, and Collings 2007)

An extensive review by (Park et al. 2008) provides examples of tandem DMA measurements. Briefly explained here, variations in experimental design have used a DMA coupled with different aerosol measurement techniques. For example, a DMA and an electrical low-pressure impactor (ELPI) coupled system – set up in series set by (Maricq, Podsiadlik, and Chase 2000) and in parallel by (Virtanen, Ristimäki, and Keskinen 2004) – determined D_m, D_ae, and ρ_eff. A DMA coupled with an optical particle counter and an aerodynamic particle sizer determined aerosol refractive index and ρ_eff (Hand and Kreidenweis 2002).

One drawback to the DMA, APM and CPMA is that these classification methods require the use of charged particles which presents complications in the analysis (Wang and Flagan 1990; Radney and Zangmeister 2016). To circumvent these issues, the aerodynamic aerosol classifier (AAC) was developed in 2013 (Tavakoli and Olfert 2013) and instead uses the response of the particle to an applied centrifugal force and counteracting drag force to determine the aerodynamic diameter (D_ae). The AAC was first demonstrated in a tandem AAC-DMA configuration by (Tavakoli and Olfert 2014) to measure the m, ρ_eff, D_fm and χ_eff of liquid dioctyl sebacate droplets and fresh soot. Utilizing this triad of instrumentation in combination allows for the determination of the dynamic shape factor (χ) and particle density (ρ_p).

The pairwise collection of an AAC, DMA and APM (i.e., AAC-DMA, AAC-APM and DMA-APM) allows for independent determinations of D_ae, D_m and m and the derivation ρ_eff and χ, see Figure 1. Since the AAC, DMA and APM use different methods for aerosol classification (i.e., relaxation time, electrical mobility and mass-to-charge ratio), tandem combinations may result in variations in measured properties. In this study, multiple combinations of tandem measurements (AAC-DMA, AAC-APM and DMA-APM) are utilized to determine m, ρ_eff and χ_eff for solid, nearly spherical particles composed of ammonium sulfate. The results from these three independent measurements are compared and discussed. Utilizing the triplet of instrumentation, we demonstrate the quantitative determination of χ and ρ_p for an aged black carbon mimic.

Figure 1. Schematic representation of the instruments and the corresponding measured values – mobility diameter (D_m), aerodynamic diameter (D_ae) and particle mass (m) – the tandem measurement pairs (connected by lines) and the corresponding derived values. Pairwise combination allows for determination of effective density (ρ_eff) and effective dynamic shape factor (χ_eff). The combination of all three measurements allows for the quantitative determination of the dynamic shape factor (χ) and particle density (ρ_p). The circle (black), square (red) and triangle (green) are used throughout the manuscript to denote the corresponding measurement results.

2. Materials and methods

Block diagrams of the experimental setups to perform pairwise comparisons are shown in Figure 2. The sizing instruments were compared using ammonium sulfate (AS, 1 mg mL⁻¹) and a H₂O-dispersible black carbon aerosol mimic (CB, 0.2 mg mL⁻¹) that resembles aged, collapsed soot; see Zangmeister et al. (2019) for a full description. Aerosols were generated from solution and suspension, respectively, using a constant output atomizer (TSI 3076)¹ and subsequently dried using a pair of silica gel diffusion dryers (TSI 3062). Size classification was performed by either an AAC (D_ae, Figure 2a, Cambustion) or a DMA (D_m, Figure 2b, TSI 3081 long DMA as part of a TSI 3080 electrostatic classifier).

Figure 2. Block diagram of the two experimental setups (a and b) used in this study to achieve three configurations (AAC-DMA, AAC-APM, DMA-APM). *N corresponds to soft X-ray charge neutralizer.

The D_ae-selected aerosol stream (Figure 2a) was charge neutralized using a soft X-ray source (TSI 3088) and sent to a parallel system consisting of a tandem DMA and condensation particle counter (CPC, TSI 3081 long DMA as part of a TSI 3082 electrostatic classifier with a TSI 3775 CPC) and a tandem APM (Kanomax 3602)-CPC (TSI 3775); raw data for AS can be found in Table S2 of the online supplementary information (SI). The flow through these systems was maintained at 300 cm³ min⁻¹ and 250 cm³ min⁻¹ by the CPCs. A cavity ring-down spectrometer is normally situated between the APM and CPC and receives a 50 cm³ min⁻¹ clean air backflush to prevent particle deposition on the mirrors (Radney and Zangmeister 2016). Although the CRD was in place, the CRD data was not utilized and will not be discussed in this analysis. D_ae selection spanned 150 nm to 550 nm (AS) or 150 nm to 400 nm (CB) with the aerodynamic size resolution (R_ae) of the AAC being maintained at 10 for all experiments. All reported data were generated from a single 5 min mobility diameter (D_m) or 10 min mass (m) distribution scan corresponding to 112 samples and ≈ 600 samples, respectively. Thus, the reported measurement results utilize Type B uncertainties (see SI) while reported averages utilize Type A uncertainties (1σ standard deviation across the data set).

The geometric mean mobility diameter, μ_geo (see Equation (32)(32) $$N=\mathrm{ }\sum _{i=1}^n{A}_i\mathrm{*}\hspace{0.17em}\text{exp}\hspace{0.17em}\left(\frac{-1\mathrm{*}{\left(\log D_{\mathrm{m}}-{\mu }_{\mathrm{geo},\mathrm{i}}\right)}^2}{2{\sigma }_{\mathrm{geo},i}^2}\right)$$(32) ), as a function of D_ae was determined from the AAC-DMA measurements. In the other experimental setup (Figure 2b), the aerosol stream was D_m-selected at this μ_geo and then passed to a parallel system consisting of a tandem DMA-CPC (i.e., scanning mobility particle sizer, SMPS) and a tandem APM-CPC; raw data for AS can be found in Table S3 of the SI. DMA and APM measurements used sheath:aerosol flow of 10:1 and a classification parameter (λ_c) of 0.32, respectively (Ehara et al. 1996).

The operational parameters of the AAC and DMA were chosen to maintain an aerodynamic size resolution (R_ae, see Equation (24)(24) $$\frac{1}{R_{\mathrm{ae}}}=\frac{\mathrm{\Delta }{D}_{\mathrm{ae}}}{D_{\mathrm{ae}}}$$(24) ) and mobility bandwidth (ΔZ_p/Z_p, see Equation (25)(25) $$\frac{\mathrm{\Delta }{Z}_{\mathrm{p}}}{Z_{\mathrm{p}}}=\frac{Q_{\mathrm{aero}}}{Q_{\mathrm{sh}}}$$(25) ) of 0.1, respectively. Unlike the DMA and AAC, the APM transfer function is an asymmetric trapezoid and the peak transmission efficiency is λ_c-dependent (Kuwata 2015). For the APM with λ_c = 0.32, Δm/m in the negative and positive directions is to -0.24 and 0.27, respectively; see Figure 4. However, since m scales with D³, the larger Δm/m is comparable to the R_ae and ΔZ_p/Z_p of 0.1; see Section 3.4 for a complete discussion of the transfer functions.

The DMAs were calibrated using polystyrene latex spheres atomized from water prior to initiation of the described experiments. Additional details in the calibration and characterization of the DMA and APM are described in Radney and Zangmeister (2016); results shown in Figure S1 of the SI. AAC performance was verified by the manufacturer at the beginning of this study and was reevaluated by the manufacturer at the end of data collection.

3. Theory

The fundamental theory and operational parameters for the AAC, DMA and APM are well-known; see Tavakoli and Olfert (2013), Knutson and Whitby (1975) and Ehara et al. (1996), respectively. Here we give an overview of the relevant theory to highlight the nuances between instruments that will be relevant to the presented results and discussion. As shown in Figure 3, particle classification by the AAC, DMA and APM rely on a combination of the non-contact forces: centrifugal (F_cen), drag (F_drag), and/or electrostatic (F_elec). In addition to these forces (black arrows), particle streamlines (gray dotted lines) and flows have been included; the centrifugal particle mass analyzer (CPMA) has also been included in Figure 3d for comparison to the APM. All figures are drawn with assuming the radial direction is upwards and flows progress from left to right.

Figure 3. Forces (F), flows (Q), axes of rotation (ω) and particle trajectories (dotted gray lines) in the (a) aerodynamic aerosol classifier (AAC), (b) differential mobility analyzer (DMA), (c) the aerosol particle mass analyzer (APM) and (d) the centrifugal particle mass analyzer (CPMA). All figures are oriented with the radial (r) and transversal (z) axes shown in the center of the figure. Subscripts: aerosol (a), sheath (sh), exhaust (exh), sample (sa), electrical (elec), centrifugal (cen).

Both the aerodynamic diameter (D_ae) and the mobility diameter (D_m) quantified by the AAC and the DMA, respectively, are defined relative to a spherical particle. Spherical particles are often assumed, although not necessarily accurate. For comparing sizes, the volume equivalent diameter (D_ve) (1) $$m=\mathrm{ }{\rho }_{\mathrm{p}}\frac{\pi }{6}{D}_{\mathrm{ve}}^3$$(1) can be used where m is the particle mass and ρ_p is the particle density, including air voids. For homogeneous particles without air voids, ρ_p = ρ_bulk.

3.1. AAC

Any particle moving at a constant velocity (v) experiences equal applied and drag forces (2) $$F_{\mathrm{app}}=F_{\mathrm{drag}}$$(2)

where for a sphere in any flow regime (e.g., free-molecular, transition or continuum) (Kulkarni, Baron, and Willeke 2011) (3) $$F_{\mathrm{drag}}=\frac{v}{B_{\mathrm{sp}}}$$(3) v and μ are the particle’s velocity and the gas viscosity and B_sp is the mechanical mobility of a sphere (4) $$B_{\mathrm{sp}}=\frac{C_{\mathrm{c}}\left(D_{\mathrm{sp}}\right)}{3\pi \mu D_{\mathrm{sp}}}$$(4)

The drag force experienced by non-spherical particles is greater than that experienced by a volume equivalent sphere with the same velocity and flow regime effectively reducing B. The dynamic shape factor (χ) represents the ratio of B for a non-spherical particle (B_non) to that of the volume equivalent sphere (B_ve) (Kasper 1982) (5) $$\chi =\frac{B_{\mathrm{ve}}}{B_{\mathrm{non}}}=\frac{F_{\mathrm{non}}}{F_{\mathrm{ve}}}\frac{v_{\mathrm{ve}}}{v_{\mathrm{non}}}$$(5)

The Cunningham slip correction factor (C_c(D)) has been included to account for decreased drag in the free-molecular and transition regimes relative to the continuum regime. The generic form of the Cunningham slip correction factor C_c(D) (Kulkarni et al. 2011) is (6) $$C_{\mathrm{c}}\left(D\right)=1+\frac{2{\lambda }_{\mathrm{g}}}{D}\left[1.142+0.558e^{\left(\raisebox{1ex}{$-0.999D$}\!\left/ \!\raisebox{-1ex}{$2{\lambda }_{\mathrm{g}}$}\right.\right)}\right]$$(6) where λ_g is the mean free path of the gas at the temperature (T) and pressure (P) of the measurement. For air at T = 298.15 K and P = 101.325 kPa: μ = 1.837 × 10⁻⁵kg m⁻¹ s⁻¹ and λ_g = 67.9 nm.

The rotating annular region of the AAC applies a centrifugal force (F_cen) (7) $$F_{\mathrm{cen}}=m{\omega }^{2}r$$(7) where r and ω are the radial position of the particle and rotation speed, respectively. Thus, (8) $$\frac{v}{B_{\mathrm{sp}}}=m{\omega }^{2}r$$(8) with the particle relaxation time (τ) defined as (Hinds 1999) (9) $$\tau \equiv m{B}_{\mathrm{sp}}$$(9)

Combining Equations (8)(8) $$\frac{v}{B_{\mathrm{sp}}}=m{\omega }^{2}r$$(8) and (9)(9) $$\tau \equiv m{B}_{\mathrm{sp}}$$(9) (10) $$\tau =\frac{v}{{\omega }^{2}r}$$(10)

allowing τ to be directly measured (11) $$\tau =\frac{Q_{\mathrm{sh}}+Q_{\mathrm{exh}}}{\pi {\omega }^2{\left(r_1+r_2\right)}^{2}L}$$(11) where Q_sh, Q_exh, r₁, r₂ and L are the sheath flow, exhaust flow, inner classifier radius (56 mm), outer classifier radius (60 mm) and the classifier length (206 mm), respectively. If operating under balanced flows, then Q_sh = Q_exh and Q_a = Q_s (the aerosol and sample flows, respectively) (see Figure 2a) and the numerator of Equation (11)(11) $$\tau =\frac{Q_{\mathrm{sh}}+Q_{\mathrm{exh}}}{\pi {\omega }^2{\left(r_1+r_2\right)}^{2}L}$$(11) simplifies to 2Q_sh.

The aerodynamic diameter (D_ae) is defined as the diameter of a sphere with standard density (ρ₀ = 1 g cm⁻³) and the same velocity as that being measured. Since m = ρV (12) $$\tau ={\rho }_p\frac{\pi }{6}{D}_{\mathrm{sp}}^3{B}_{\mathrm{sp}}={\rho }_0\frac{\pi }{6}{D}_{\mathrm{ae}}^3\chi B_{\mathrm{ae}}$$(12) which rearranges to (13) $$D_{\mathrm{ae}}=D_{\mathrm{sp}}\sqrt{\frac{{\rho }_{\mathrm{p}}{C}_{\mathrm{c}}\left(D_{\mathrm{sp}}\right)}{\chi {\rho }_0{C}_{\mathrm{c}}\left(D_{\mathrm{ae}}\right)}}$$(13)

and it follows that (14) $$\tau =\frac{{\rho }_0{D}_{\mathrm{ae}}^2{C}_{\mathrm{c}}\left(D_{\mathrm{ae}}\right)}{18\mu }$$(14)

Note that determination of D_ae requires that either ρ_p and χ are known or assumed thus making D_ae an estimated parameter; χ = 1 is often assumed in Equation (14)(14) $$\tau =\frac{{\rho }_0{D}_{\mathrm{ae}}^2{C}_{\mathrm{c}}\left(D_{\mathrm{ae}}\right)}{18\mu }$$(14) to allow this underdetermined system to be solved. An effective dynamic shape factor (χ_eff) is then determined from Equation (13)(13) $$D_{\mathrm{ae}}=D_{\mathrm{sp}}\sqrt{\frac{{\rho }_{\mathrm{p}}{C}_{\mathrm{c}}\left(D_{\mathrm{sp}}\right)}{\chi {\rho }_0{C}_{\mathrm{c}}\left(D_{\mathrm{ae}}\right)}}$$(13) as in (Tavakoli and Olfert 2013; Tavakoli, Symonds, and Olfert 2014; Kazemimanesh et al. 2019). In another instance, (Tavakoli and Olfert 2014) did not need to explicitly account for χ since only D_ae and D_m, not D_ve, were compared. See discussion below.

3.2. DMA

In the DMA, an electric field is applied between two, usually concentric, electrodes (see Figure 2b) imparting and electrostatic force (F_elec) on the charged particles (15) $$F_{\mathrm{elec}}= \mathit{qeE}$$(15)

where q, e, and E are the net number of charges on the particle, the elementary electric charge (≈ 1.602 × 10⁻¹⁹ C) and the electric field strength, respectively. The electric mobility (Z_p) is (DeCarlo et al. 2004) (16) $$Z_{\mathrm{p}}=qe{B}_{\mathrm{m}}=\frac{qe{C}_c\left(D_{\mathrm{m}}\right)}{3\pi \mu D_{\mathrm{m}}}$$(16) where D_m is defined as the diameter of a sphere with the same electrical mobility as that being measured. Combining Equations (5)(5) $$\chi =\frac{B_{\mathrm{ve}}}{B_{\mathrm{non}}}=\frac{F_{\mathrm{non}}}{F_{\mathrm{ve}}}\frac{v_{\mathrm{ve}}}{v_{\mathrm{non}}}$$(5) and (16)(16) $$Z_{\mathrm{p}}=qe{B}_{\mathrm{m}}=\frac{qe{C}_c\left(D_{\mathrm{m}}\right)}{3\pi \mu D_{\mathrm{m}}}$$(16) , (17) $$\frac{D_{\mathrm{m}}}{C_{\mathrm{c}}\left(D_{\mathrm{m}}\right)}=\frac{{\chi D}_{\text{ve }}}{C_{\mathrm{c}}\left(D_{\mathrm{ve}}\right)}$$(17) and D_ve does not necessarily equal D_sp. DMA measurements typically assume a single net charge is present on the particle. Z_p can be related to the physical dimensions of the classifier by (18) $$Z_{\mathrm{p}}=\frac{Q_{\mathrm{sh}}}{2\pi VL}\ln \left(\raisebox{1ex}{$r_2$}\!\left/ \!\raisebox{-1ex}{$r_1$}\right.\right)$$(18) where r₁ = 0.937 cm, r₂ = 1.961 cm and L = 44.369 cm and V is the applied voltage.

3.3. APM

The aerosol particle mass analyzer (APM) is essentially the combination DMA and AAC that is capable of directly measuring particle mass. The classifier rotates about an axis generating F_cen (Equation (7)(7) $$F_{\mathrm{cen}}=m{\omega }^{2}r$$(7) ) while an electric potential is applied to the inner electrode generating F_elec (Equation (15)(15) $$F_{\mathrm{elec}}= \mathit{qeE}$$(15) ); see Figure 2c. Thus, (19) $$F_{\mathrm{cen}}=F_{\mathrm{elec}}$$(19)

and (20) $$m{\omega }^{2}r=\mathit{qeE}=\frac{\mathit{qeV}}{r\ln \left(\raisebox{1ex}{$r_2$}\!\left/ \!\raisebox{-1ex}{$r_1$}\right.\right)}$$(20) where r₁ = 24 mm and r₂ = 25 mm. As can be seen from Equation (20)(20) $$m{\omega }^{2}r=\mathit{qeE}=\frac{\mathit{qeV}}{r\ln \left(\raisebox{1ex}{$r_2$}\!\left/ \!\raisebox{-1ex}{$r_1$}\right.\right)}$$(20) , the APM classifies particles based upon their effective mass (m_eff) where (21) $$m_{\mathrm{eff}}=\frac{m}{q}$$(21)

During APM classification, the particles of interest will have no net radial velocity and F_drag = 0. The result is that the APM directly measures particle mass without any assumptions of particle shape (χ) but does assume a single net charge. If ρ_p is known, then D_ve can be directly calculated from Equation (1)(1) $$m=\mathrm{ }{\rho }_{\mathrm{p}}\frac{\pi }{6}{D}_{\mathrm{ve}}^3$$(1) , otherwise, in combination with a DMA, the effective density ρ_eff is calculated as (22) $${\rho }_{\mathrm{eff}=}\frac{m}{\left(\frac{\pi }{6}\right)D_{\mathrm{m}}^3}$$(22)

It is worth mentioning that the Cambustion centrifugal particle mass analyzer (CPMA) (see Figure 2d) operates similarly to the Kanomax APM except that the inner electrode rotates slightly faster than the outer electrode causing F_cen to decrease with radius. This causes particles to exhibit a stable flow for a larger fraction of the operating area impacting the transfer function. We direct the reader to (Olfert and Collings 2005) and (Sipkens, Olfert, and Rogak 2020b) for an in-depth comparison of the APM and CPMA.

3.4. Transfer functions

Thus far, the transfer functions of the AAC, DMA and APM have been treated as Dirac delta functions. The actual transfer functions have finite widths that are a function of the operating parameters. Since the AAC, DMA and APM utilize different mechanisms for particle classification (relaxation time, electrical mobility and mass-to-charge ratio) these resolutions are often reported utilizing different metrics: relaxation time (R_τ) or aerodynamic size (R_ae) resolution, sheath:aerosol flow and the classification parameter (λ_c), respectively. Here, we relate these resolution metrics to the height and full width of the transfer functions at baseline.

For the AAC, the relaxation time resolution (R_τ) is defined based upon the ratio of Q_sh to Q_aero and assumes that these flows are balanced, laminar and constant: (23) $$\frac{1}{R_{\tau }}=\frac{\mathrm{\Delta }\tau }{\tau }=\frac{Q_{\mathrm{aero}}}{Q_{\mathrm{sh}}}$$(23)

The AAC transfer function is an isosceles triangle with respect to relaxation time. The fraction of particles transmitted at τ, t(τ) = 1.0, and drops to 0 at τ ± R_τ (Tavakoli and Olfert 2013). Rather than utilizing R_τ, the AAC resolution can be defined based upon the aerodynamic diameter resolution (R_ae) (24) $$\frac{1}{R_{\mathrm{ae}}}=\frac{\mathrm{\Delta }{D}_{\mathrm{ae}}}{D_{\mathrm{ae}}}$$(24)

For constant R_ae, the rotation speed and sheath flow of the AAC are a function of the classified particles. Presently, R_ae = 10 with Q_sh and ω adjusted accordingly by the instrument. The full width of the transfer function is two times R_ae and is 0.2 (R_ae = 10).

For the DMA – assuming laminar flow and that the particles are non-diffusing – the transfer function is an isosceles triangle with respect to the electrical mobility (Z_p). The fraction of particles transmitted at Z_p, t(Z_p) = 1.0 (Knutson and Whitby 1975; Hagwood, Sivathanu, and MulHolland 1999; Kuwata 2015) and drops to 0 at Z_p ± ΔZ_p where (25) $$\frac{\mathrm{\Delta }{Z}_{\mathrm{p}}}{Z_{\mathrm{p}}}=\frac{Q_{\mathrm{aero}}}{Q_{\mathrm{sh}}}$$(25)

The full width of the transfer function is two times ΔZ_p/Z_p and for the present measurements is 0.2 (Q_sh/Q_aero = 10). The Z_p dependence of the DMA transfer function causes it to be asymmetric with respect to D_m.

The transfer function of the APM – assuming uniform laminar flow – is an asymmetric trapezoid with a center at m_eff (Lall et al. 2008, 2009; Kuwata 2015). The APM classification parameter (λ_c) is (26) $${\lambda }_{\mathrm{c}}=\frac{2\tau {\omega }^{2}L}{v}$$(26) and represents the ratio of the axial (L/v) and radial (1/2τω²) transversal times of an uncharged particle. Following (Kuwata 2015), the width of the transfer function at baseline in the positive and negative directions (Δm⁺/m_s and Δm^-/m_s, respectively, or Δm^±/m_s collectively) is (27) $$\frac{\mathrm{\Delta }{m}^{\pm }}{m_{\mathrm{eff}}}=-2\mathrm{l}\mathrm{n}\left[1\mp \left(\frac{\delta }{r_{\mathrm{c}}}\right)\coth \left(\frac{{\lambda }_{\mathrm{c}}}{2}\right)\right]$$(27) where (28) $$\delta =\frac{\left(r_1-r_2\right)}{2}$$(28)

and the center of the classification region (r_c) (29) $$r_{\mathrm{c}}=\frac{\left(r_1+r_2\right)}{2}$$(29) r₁ and r₂ are the radius of the inner (24 mm) and outer (25 mm) electrodes, respectively. The transmission probability is (30) $$t\left(m\right)=e^{-{\lambda }_{\mathrm{c}}}$$(30)

spanning from (31) $$\frac{\Delta m_2^{\pm }}{m_{\mathrm{eff}}}=-2\mathrm{l}\mathrm{n}\left(1\mp \frac{\delta }{r_{\mathrm{c}}}\right)$$(31)

Presently, λ_c = 0.32 corresponding to Δm^-/m_eff = 0.242 and Δm⁺/m_eff = 0.275 and t(m_eff) = 0.73.

The DMA, AAC and APM transfer functions in their native measurands (Z_p, D_ae and m) are plotted in Figures 4a–c, respectively. The combined values transformed to D_ve are shown in Figure 4d utilizing the operational parameters outlined above. The absolute widths of the transfer functions will scale with size, so these values should be considered representative for this D_ve. For these calculations, a spherical AS particle with a physical diameter of 250 nm and bulk density was assumed; i.e., D_p = D_m = D_ve = 250 nm, ρ_p = ρ_eff = ρ_bulk = 1.77 g cm⁻³, χ = 1, m_p = 14.4 fg and D_ae = 356 nm.

Figure 4. Calculated transfer functions for the (a) DMA (solid), (b) AAC (dashed) and (c) APM (dotted) in their native units of electrical mobility (Z_p), aerodynamic diameter (D_ae), and mass (m). (d) All transfer functions converted to volume equivalent diameter (D_ve) for a spherical AS assuming a physical diameter of 250 nm and bulk density; see discussion in text.

Note, the transfer function calculated in Figure 4c is for the APM and not the CPMA. The CPMA transfer function has a similar trapezoidal shape but higher t(m) = 1 independent of λ_c; see Olfert and Collings (2005) and Sipkens et al. (2020b) for detailed comparisons of the APM and CPMA transfer functions.

3.5. Data inversions

Scanning measurements by the AAC, DMA or APM are often performed and the data must be inverted in order to determine the underlying particle distributions; e.g., Mai and Flagan (2018) and Stolzenburg (2018) for the DMA and Rawat et al. (2016) and Sipkens Sipkens, Olfert, and Rogak (2020a) for the DMA-APM. However, in the present work, data inversions were not performed since the parameters of interest (D_ae, D_m and m for the AAC, DMA and APM, respectively) could be determined directly from the instrumental setpoint (D_ae and D_m) or from fitting the distributions of the scanned data (D_m and m). We assume that the aerosol concentration is constant over the width of the transfer functions (Stolzenburg and McMurry 2018) but expect this assumption to have little impact on our measurements since: (1) the geometric or arithmetic means are the quantity of interest and (2) these deviations are small relative to the instrumentally imposed measurement resolution.

3.6. Multiple charging

All classification methods that are charge dependent suffer from multiple charging issues (|q| > 1) and this is especially true for the DMA and APM. Prior to classification by these instruments, the aerosol stream is passed through a charge neutralizer (*N in Figure 2) to impart a known bipolar charge distribution to the particles (Wiedensohler and Fissan 1988; Tigges et al. 2015). The q of particles exiting the charge neutralizer is a strong function of particle size, shape (Rogak, Flagan, and Nguyen 1993), neutralizer age and type (Tigges et al. 2015), and to a lesser extent particle morphology (Covert, Wiedensohler, and Russell 1997).

To demonstrate these multiple charging artifacts, we performed tandem DMA (TDMA) measurements whereby particles exiting an upstream DMA are passed through a second neutralizer and DMA and the corresponding size distribution was measured; see Figure 2b. A representative TDMA size distribution is shown in Figure 5a for AS aerosol with a nominal D_ae ≈ 250 nm (D_m = 169 nm). The notation q → q’ in Figure 4a denotes the charge of particles exiting the first and second DMAs, respectively. The size distribution exhibits multiple peaks beyond just that selected by the upstream DMA (D_m = 169 nm); in the case of D_m = 169 nm (green line), the peak is composed primarily of particles with q = q′ = +1 with smaller fractions of q = q′ = +2 and +3 also present (Mamakos 2016). In addition, q → q′ of +2 → +1, +3 → +1, +1 → +2, +1 → +3, +2 → +3 and +3 → +2 are observed at D_m = 265.2 nm, 352.9 nm, 110.1 nm, 87.33 nm, ≈ 217 nm and ≈ 131 nm, respectively. To determine D_m, the mobility distributions were fit utilizing multiple log-normal distributions (32) $$N=\mathrm{ }\sum _{i=1}^n{A}_i\mathrm{*}\hspace{0.17em}\text{exp}\hspace{0.17em}\left(\frac{-1\mathrm{*}{\left(\log D_{\mathrm{m}}-{\mu }_{\mathrm{geo},\mathrm{i}}\right)}^2}{2{\sigma }_{\mathrm{geo},i}^2}\right)$$(32) and A, μ_geo and σ_geo are the peak amplitude, the geometric mean diameter and the geometric standard deviation. For consistency, we will continue to refer to fitted μ_geo as the D_m of the distribution. The summation and the i subscripts denote that all peaks with N > 7.5 × 10⁸ m⁻³ were included in the fit. Other higher order multiples could also be present but unresolved – e.g., +4 → +2, +6 → +3, etc. (Mamakos 2016) – since the soft X-ray neutralizer produces a greater fraction of q > +1 than the radioactive neutralizers (Tigges et al. 2015). These redistributed charges are only present in the TDMA measurements and only apply to the illustrative data presented here. Even if a significant number of higher order charges are present, they are not expected to impact the determination of D_m.

Figure 5. (a) Tandem DMA number density (N_TDMA) as a function of mobility diameter for AS particles with an aerodynamic diameter (D_ae) ≈ 250 nm. Particle net charge (q) is denoted as q → q′ where q and q′ correspond to the q in the first and second DMA, respectively. (b) DMA-APM number density (N_APM) as a function of mass (m). The net charge +1 (green), +2 (cyan) and +3 (blue) on the particles is shown.

To determine the average particle mass, the distribution of N as a function of m was measured by the APM and were fit utilizing multiple Gaussian distributions (33) $$N={\sum }_{q=1}^n{A}_q\mathrm{*}\exp \left(\frac{{-\left(m-m_{\mathrm{eff},q}\right)}^2}{2{\sigma }_{\mathrm{eff},q}^2}\right)$$(33) where A, m and σ correspond to the peak amplitude, the mass and the width (i.e., standard deviation) of the distribution. Similar to D_m (Equation (32)(32) $$N=\mathrm{ }\sum _{i=1}^n{A}_i\mathrm{*}\hspace{0.17em}\text{exp}\hspace{0.17em}\left(\frac{-1\mathrm{*}{\left(\log D_{\mathrm{m}}-{\mu }_{\mathrm{geo},\mathrm{i}}\right)}^2}{2{\sigma }_{\mathrm{geo},i}^2}\right)$$(32) ), and for consistency throughout, we will continue to refer to the average mass of the distribution (m_eff,q) as m. The summations and q have been included to denote that all peaks with q ≥ +1 were included in the fit; see Figure 5b. In instances where particles are non-spherical with low effective densities, even this approach has its limitations (Radney et al. 2014).

When performing DMA-APM measurements, the particles are not sent through a second charge neutralizer prior to the APM. Thus, all particles exiting the DMA having a common electrical mobility with physical diameters (and hence masses) that scale with charge as shown by the +2 → +1 (cyan) and +3 → +1 (blue) traces in Figure 5a. These particles bearing larger charges can also be seen in the mass distributions of N as shown in Figure 5b.

Separation of particles by an AAC, versus a DMA, poses slightly different challenges when combined with a downstream DMA or APM (Figure 2a). The AAC separates particles based upon their relaxation time to an applied centrifugal force and is charge independent. However, upon passing from the AAC to either a DMA or APM, the particles must pass through a charge neutralizer in order to reach an equilibrium bipolar charge distribution independent of the incoming charge state (Wiedensohler and Fissan 1988). This causes the size distribution to exhibit multiple peaks where all particles have a common D_ae but a q-dependent D_m; see Figure 6a for AAC-DMA measurements of AS with D_ae = 350 nm. In the AAC-APM, versus the DMA-APM, the contributions of particles bearing q > +1 cannot necessarily be isolated as seen in Figure 6b.

Figure 6. (a) AAC-DMA number density (N_SMPS) as a function of mobility diameter for AS particles with aerodynamic diameter (D_ae) = 350 nm. (b) AAC-APM number density (N_APM). The net charge +1 (green), +2 (cyan) and +3 (blue) on the particles is shown.

For the purpose of comparing the tandem measurements (AAC-DMA, AAC-APM and DMA-APM) the quantities of interest will derive from the geometric and arithmetic means of the measured size and mass distributions by the DMA and APM, respectively. As a result, the impact of multiple charging on the reported results is only expected to be significant for smaller particles where overlap between successive charges increases. Non-spherical particles with low effective densities, such as fresh soot, would exhibit similar problems but are not considered here; e.g., Radney et al. (2014) and Tavakoli and Olfert (2014). The smallest AS particles measured presently had D_m ≈ 100 nm with ρ_p = 1.77 g cm⁻³ while for CB the smallest D_m ≈ 150 nm and ρ_p = 1.00 g cm⁻³ so the multiple charging effects are resolvable.

3.7. Comparing tandem measurements

A fundamental objective of this article is to compare the measured and derived parameters from each of the tandem techniques (AAC-DMA, AAC-APM and DMA-APM); see Figures 1 and 2. In the case of the AAC-DMA, τ and Z_p are measured (Equations (11)(11) $$\tau =\frac{Q_{\mathrm{sh}}+Q_{\mathrm{exh}}}{\pi {\omega }^2{\left(r_1+r_2\right)}^{2}L}$$(11) and (18)(18) $$Z_{\mathrm{p}}=\frac{Q_{\mathrm{sh}}}{2\pi VL}\ln \left(\raisebox{1ex}{$r_2$}\!\left/ \!\raisebox{-1ex}{$r_1$}\right.\right)$$(18) , respectively) from which B (Equation (16)(16) $$Z_{\mathrm{p}}=qe{B}_{\mathrm{m}}=\frac{qe{C}_c\left(D_{\mathrm{m}}\right)}{3\pi \mu D_{\mathrm{m}}}$$(16) ), and hence m (Equation (9)(9) $$\tau \equiv m{B}_{\mathrm{sp}}$$(9) ) can be calculated. For the AAC-APM, τ and m are measured (Equations (11)(11) $$\tau =\frac{Q_{\mathrm{sh}}+Q_{\mathrm{exh}}}{\pi {\omega }^2{\left(r_1+r_2\right)}^{2}L}$$(11) and (20)(20) $$m{\omega }^{2}r=\mathit{qeE}=\frac{\mathit{qeV}}{r\ln \left(\raisebox{1ex}{$r_2$}\!\left/ \!\raisebox{-1ex}{$r_1$}\right.\right)}$$(20) , respectively), from which B (Equation (9)(9) $$\tau \equiv m{B}_{\mathrm{sp}}$$(9) ), and hence D_m (Equation (16)(16) $$Z_{\mathrm{p}}=qe{B}_{\mathrm{m}}=\frac{qe{C}_c\left(D_{\mathrm{m}}\right)}{3\pi \mu D_{\mathrm{m}}}$$(16) ) can be calculated. For the DMA-APM, Z_p and m are measured (Equations (17)(17) $$\frac{D_{\mathrm{m}}}{C_{\mathrm{c}}\left(D_{\mathrm{m}}\right)}=\frac{{\chi D}_{\text{ve }}}{C_{\mathrm{c}}\left(D_{\mathrm{ve}}\right)}$$(17) and (20)(20) $$m{\omega }^{2}r=\mathit{qeE}=\frac{\mathit{qeV}}{r\ln \left(\raisebox{1ex}{$r_2$}\!\left/ \!\raisebox{-1ex}{$r_1$}\right.\right)}$$(20) , respectively) from which B, and hence τ (Equation (9)(9) $$\tau \equiv m{B}_{\mathrm{sp}}$$(9) ) can be determined. D_ae is determined from τ in all configurations assuming χ = 1 (Equation (14)(14) $$\tau =\frac{{\rho }_0{D}_{\mathrm{ae}}^2{C}_{\mathrm{c}}\left(D_{\mathrm{ae}}\right)}{18\mu }$$(14) ). For all tandem combinations, a ρ_p must be assumed in order to estimate D_ve (Equation (1)(1) $$m=\mathrm{ }{\rho }_{\mathrm{p}}\frac{\pi }{6}{D}_{\mathrm{ve}}^3$$(1) ) and χ_eff (Equation (5)(5) $$\chi =\frac{B_{\mathrm{ve}}}{B_{\mathrm{non}}}=\frac{F_{\mathrm{non}}}{F_{\mathrm{ve}}}\frac{v_{\mathrm{ve}}}{v_{\mathrm{non}}}$$(5) ).

4. Results and discussion

4.1. Ammonium sulfate

In principle, measured pairwise combinations of τ (D_ae), Z_p (D_m) and/or m by an AAC, DMA and/or APM, respectively, can be used to derive the third quantity (as shown in Figure 1). In one configuration, particles were classified by D_ae with the AAC and D_m and m_p were measured in parallel (Figure 2a). For direct comparability between measurements, in the second configuration (Figure 2b), particles were classified by the DMA using the measured D_m from the AAC-DMA. As a result, the D_m data from the DMA-APM has not been included in the D_m comparisons (see Figures 7a).

Figure 7. Comparison of three different measurements – AAC-DMA (circles, black), AAC-APM (squares, red), and DMA-APM (triangles, green) – of ammonium sulfate (AS) aerosol spanning D_ae from 150 nm to 550 nm. (a) Mobility diameter (D_m) determined from AAC-DMA measurement and AAC-APM calculation; (b) D_m % deviation; (c) particle mass (m); (d) m % deviation; (e) effective density (ρ_eff); (f) ρ_eff % deviation; (g) effective shape factor (χ_eff); (h) χ_eff % deviation.

In the case of AS, the calculated D_m (AAC-APM, red squares) were consistently smaller than the measured D_m (AAC-DMA, black circles), see Figure 7a, with an average deviation of (-4 ± 1)% (Figure 7b). The relative deviations between measurements (i) are calculated as (34) $$\frac{(i_a\mathrm{ }–\mathrm{ }{i}_b)}{\mathrm{ }{i}_b}\times 100\mathrm{\%}$$(34)

The relative deviations of Figure 7b were calculated with a representing the AAC-APM and b the AAC-DMA. For all other calculations of relative deviation (Figure 7d, f, and h), a represents either the AAC-DMA or AAC-APM and b the DMA-APM.

The AAC-DMA measures τ and Z_p from which the D_ae (Equation (14)(14) $$\tau =\frac{{\rho }_0{D}_{\mathrm{ae}}^2{C}_{\mathrm{c}}\left(D_{\mathrm{ae}}\right)}{18\mu }$$(14) ) and D_m (Equation (16)(16) $$Z_{\mathrm{p}}=qe{B}_{\mathrm{m}}=\frac{qe{C}_c\left(D_{\mathrm{m}}\right)}{3\pi \mu D_{\mathrm{m}}}$$(16) ) are derived. Similar to the investigations of (Tavakoli and Olfert 2013; Tavakoli et al. 2014; Kazemimanesh et al. 2019), we assume that χ = 1 in Equation (14)(14) $$\tau =\frac{{\rho }_0{D}_{\mathrm{ae}}^2{C}_{\mathrm{c}}\left(D_{\mathrm{ae}}\right)}{18\mu }$$(14) allowing for D_ae to be calculated; χ_eff is later calculated from D_ve. The m calculated from the AAC-DMA can be compared to the m directly measured by the AAC-APM and DMA-APM; see Figure 7c. In general, the AAC-DMA tends to report the highest m of all measurements. The average m deviation between the DMA-APM and the AAC-DMA and the AAC-APM are (9 ± 5)% and (4 ± 5)% for AS, respectively. All measures of m agree within 15%.

Particle effective density (ρ_eff) is an important parameter that can serve as a metric of particle morphology and may be used to convert size distributions to mass distributions or from mobility distributions to aerodynamic distributions (Johnson et al. 2018); the μ_geo of the D_m fits were utilized in the calculation of ρ_eff, see Equations (22)(22) $${\rho }_{\mathrm{eff}=}\frac{m}{\left(\frac{\pi }{6}\right)D_{\mathrm{m}}^3}$$(22) and (32)(32) $$N=\mathrm{ }\sum _{i=1}^n{A}_i\mathrm{*}\hspace{0.17em}\text{exp}\hspace{0.17em}\left(\frac{-1\mathrm{*}{\left(\log D_{\mathrm{m}}-{\mu }_{\mathrm{geo},\mathrm{i}}\right)}^2}{2{\sigma }_{\mathrm{geo},i}^2}\right)$$(32) .The average ρ_eff for AS spanning 150 nm ≤ D_ae ≤ 550 nm are (1.62 ± 0.02) g cm⁻³, (1.75 ± 0.01) g cm⁻³ and (1.91 ± 0.04) g cm⁻³ for the DMA-APM, AAC-DMA and AAC-APM, respectively. The % deviations are (8 ± 2)% and (18 ± 3)% for the AAC-DMA and AAC-APM versus the DMA-APM, respectively (Figure 7f). Notably, the ρ_eff determined from the AAC-APM measurements are not physically reasonable considering the bulk density of AS is 1.77 g cm⁻³ and these particles are not expected to be perfectly spherical and may contain voids (Zelenyuk, Cai, and Imre 2006).

The dynamic shape factor (χ) is another useful metric for quantifying particle morphology. It is defined as the ratio of the drag force experienced by a non-spherical particle to the drag force experienced by a volume-equivalent spherical particle with the same velocity and flow regime with values of 1 for perfect spheres and ≫ 1 for lacey aggregates. In the case of AS, the DMA-APM, AAC-DMA and AAC-APM yielded χ_eff of 1.05 ± 0.01, 1.006 ± 0.005, 0.97 ± 0.03, respectively; we refer to these as effective dynamic shape factors (χ_eff) because a particle density (ρ_p) – here ρ_p = ρ_bulk = 1.77 g cm⁻³ – must be assumed. This is different than the dynamic shape factor (χ) determined for CB below from the triplet of instruments where ρ_p is not assumed. The deviations are (-4 ± 1)% and (-8 ± 1)% for the AAC-DMA and AAC-APM versus the DMA-APM, respectively. As with ρ_eff, the AAC-APM measurements are not physically reasonable (χ < 1), even though all measurements are within 8% of each other.

4.2. Data harmonization

The governing behavior of the classifiers (AAC, DMA and APM) and associated mathematical relationships are highly interconnected due to their similarity (see Figure 3). On average there is an 8% deviation in ρ_eff between the AAC-DMA (1.75 ± 0.01) g cm⁻³, and the DMA-APM (1.62 ± 0.02) g cm⁻³ and an 18% deviation between the DMA-APM and the AAC-APM (1.91 ± 0.04) g cm⁻³. If instead ρ_eff was calculated with the D_m from the AAC-DMA and m_p from the AAC-APM, ρ_eff = (1.66 ± 0.1) g cm⁻³ and the deviation decreases to 2.7%; within measurement uncertainty (see Table S1 of the SI for uncertainties). This indicates that the m calculated from the AAC-DMA and D_m calculated from the AAC-APM are responsible for the deviations.

The relaxation time (τ = mB, Equation (9)(9) $$\tau \equiv m{B}_{\mathrm{sp}}$$(9) ) assumes sphericity. However, the mass that of interest is for a non-spherical particle (AS) requiring that χ be included in Equation (9)(9) $$\tau \equiv m{B}_{\mathrm{sp}}$$(9) ; i.e. (35) $$m=\frac{\tau }{\chi B_{\mathrm{m}}}$$(35)

Utilizing χ_eff from the DMA-APM to calculate m in Equation (35)(35) $$m=\frac{\tau }{\chi B_{\mathrm{m}}}$$(35) decreases the average ρ_eff from (1.75 ± 0.01) g cm⁻³ to (1.67 ± 0.02) g cm⁻³ with a deviation of 3.4%, nearly within the calculated uncertainty of 2.9% and 3.2% for the AAC-DMA and DMA-APM, respectively. This also implies that the D_ae calculated from τ should also be decreased by ≈ √τ (Equation (14)(14) $$\tau =\frac{{\rho }_0{D}_{\mathrm{ae}}^2{C}_{\mathrm{c}}\left(D_{\mathrm{ae}}\right)}{18\mu }$$(14) ).

Like τ, D_m assumes sphericity, so the actual particle mobility will be smaller by a factor of χ. Utilizing χ_eff from the DMA-APM to calculate D_m for the AAC-APM decreases ρ_eff from (1.91 ± 0.04) g cm⁻³ to (1.62 ± 0.02) g cm⁻³ and the deviation is < 1%.

The observations presented above demonstrate that data harmonization between the DMA-APM, AAC-DMA and AAC-APM requires the use of χ since it is the conversion factor between sphericity and non-sphericity. To a lesser extent, data harmonization could also be impacted by: (1) the assumed physical and thermodynamic parameters (μ = 18.3245 × 10⁻⁶Pa s and λ_g= 68.29 nm at T = 295.61 K and P = 99.6 kPa) and (2) the Cunningham slip correction factor which is dependent upon both the surface roughness and the physical state (solid or liquid) of the particle (Allen and Raabe 1985) and have uncertainties on the order of 1.2% (Tavakoli and Olfert 2014) and 2.1% (Allen and Raabe 1985), respectively.

Tavakoli and Olfert (2014) measured the ρ_eff of dioctyl sebacate (DOS), an organic liquid, using the AAC-DMA and obtained ρ_eff that agreed very well with ρ_bulk: (0.903 ± 0.090) g cm⁻³ versus (0.913 ± 0.003) g cm⁻³, respectively, a -1.1% difference. We attribute this agreement to the fact that DOS is a liquid and forms smooth, spherical, void-free particles (i.e., χ = 1 so C_c(D) is well-known) and so the analysis above was not necessary. However, it is possible that the soot masses of Tavakoli and Olfert (2014) could be in error since χ was not utilized in the determination of m, but rather calculated afterwards. For CB (see below), this distinction results in the calculated mass being ≈ 10% higher than measured.

4.3. Carbon black

In addition to AS, an aged black carbon mimic (CB) was also investigated. Unlike AS, which is a solid and nearly spherical, the CB particles consists of spherical monomers ≈ 30 nm in diameter agglomerated into a larger particle with a compacted morphology (You et al. 2016). Utilizing the combination of τ, B (from D_m) and m measured by the AAC, DMA and APM – tandem configuration of Figure 2a – χ (Equation (35)(35) $$m=\frac{\tau }{\chi B_{\mathrm{m}}}$$(35) ) and ρ_p (Equation (1)(1) $$m=\mathrm{ }{\rho }_{\mathrm{p}}\frac{\pi }{6}{D}_{\mathrm{ve}}^3$$(1) ) were be quantitatively determined to be 1.09 ± 0.03 and (1.00 ± 0.03) g cm⁻³, respectively, with a minor size dependence; see Table 1. Assuming ρ_bulk = 1.8 g cm⁻³ for CB, this implies that CB has a packing density (θ_f) of 0.55 ± 0.02 which higher than expected for soot compacted through water condensation and evaporation (Zangmeister et al. 2014).

Table 1. Measured and calculated parameters for CB.

Measured			Calculated
τ (ns)	mp (fg)	B (× 10^10 m N^−1 s^−1)	χ	Dae (nm)	Dm (nm)	ρeff (g cm^−3)	Dve (nm)	ρp (g cm^−3)
149.5	1.82	7.949	1.036	145.3	156.6	0.902	152.9	0.971
224.4	4.29	4.750	1.100	186.5	217.8	0.793	204.5	0.959
314.6	7.87	3.578	1.117	231.5	264.2	0.816	244.3	1.031
417.9	13.71	2.705	1.127	276.4	322.4	0.781	295.1	1.018
538.6	21.97	2.229	1.100	329.0	372.1	0.814	346.5	1.009
665.8	32.48	1.853	1.106	372.0	428.5	0.788	395.4	1.004

5. Summary

The AAC, DMA and APM measure the aerodynamic diameter (D_ae from τ), mobility diameter (D_m from Z_p) and mass (m). Combining these three measurements allow for the dynamic shape factor (χ) and particle density (ρ_p) to be determined quantitatively. However, the use of two out of the three quantities to define mass, size, shape and/or density is subject to uncertainty. Specifically, mass and size values determined by these three configurations vary by up to 10% causing the effective density to vary by up to 18%. More importantly, nonphysical values are sometimes reported. Further errors can arise, especially when utilizing an AAC, from assuming a dynamic shape factor (χ). Physical conditions are also important but to a lesser extent than χ; e.g., T and P affect the gas viscosity, mean free path and Cunningham slip correction factor therefore impacting D_m and D_ae. Uncertainties in the effective dynamic shape factor can easily be greater than 10% when utilizing a paired combination of instruments with the triplet circumventing these issues. Understanding these differences is required to harmonize methods, improve data agreement and enable quantitative comparability between studies.

Nomenclature
AAC	=	aerodynamic aerosol classifier
APM	=	aerosol particle mass analyzer
AS	=	ammonium sulfate
B	=	mechanical mobility (m N−1 s−1)
Cc	=	Cunningham slip correction factor
χ	=	dynamic shape factor
CB	=	carbon black
CPMA	=	centrifugal particle mass analyzer
CPC	=	condensation particle counter
D	=	diameter (nm)
DMA	=	differential mobility analyzer
Δx	=	width of x’s transfer function
e	=	elementary charge (≈ 1.602 × 10−19 C)
F	=	force (N)
L	=	length (cm)
λc	=	APM classification parameter
λg	=	mean free path (nm)
m	=	mass (fg)
μ	=	gas viscosity (kg m−1 s−1)
μgeo	=	geometric mean diameter (nm)
N	=	number density of particles (m−3)
P	=	pressure (kPa)
q	=	net charge
Q	=	flow (L min−1)
r	=	radius (cm)
R	=	resolution
ρ	=	density (g cm−3)
σ	=	distribution width
T	=	temperature (K)
τ	=	relaxation time (ns)
V	=	voltage (V)
Zp	=	electrical mobility (m2 V−1 s−1)
ω	=	rotation speed (rotations min−1)

Subscripts
ae	=	aerodynamic
aero	=	aerosol
bulk	=	bulk
c	=	center
eff	=	effective
m	=	mobility
non	=	non-spherical
p	=	particle
s	=	sample
sh	=	sheath
sp	=	spherical
ve	=	volume equivalent
1	=	inner
2	=	outer

Additional information

Funding

This study was supported by the Greenhouse Gas Measurements Program at the National Institute of Standards and Technology and the U.S. National Science Foundation (NSF) under Grant 1708337. Its contents are solely the responsibility of the grantee and does not necessarily represent the official views of the NSF. Furthermore, the NSF does not endorse the purchase of any commercial products or services mentioned in the publication.

Notes

1 NIST Technical Disclaimer: Certain commercial equipment, instruments, or materials (or suppliers, or software, …) are identified in this paper to foster understanding. Such identification does not imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the materials or equipment identified are necessarily the best available for the purpose.