Measurement of the centrifugal particle mass analyzer transfer function

Abstract Particle mass analyzers, in particular the centrifugal particle mass analyzer (CPMA) and the aerosol particle mass analyzer (APM), have provided new possibilities for aerosol science through their ability to classify particles by their mass-to-charge ratio. The performance of the CPMA in classifying particles is characterized by a probability distribution known as a transfer function. This study shows the theoretical models of the CPMA’s transfer function that exist in the literature cannot accurately predict the CPMA’s actual performance. In this study, a tandem CPMA (TCPMA) measurement technique was used to experimentally evaluate the deviation of the actual CPMA transfer function from its idealized triangular transfer function. This deviation was measured by three factors: (i) the width factor ( ), (ii) the height factor ( ), and (iii) the mass set point agreement (i.e., the agreement between the set points of the two CPMAs, ); such that concurrent values of 1 for all three factors implied no deviation between the actual and theoretical triangular CPMA transfer functions. These factors were derived by adjusting them to fit TCPMA data with the convolution of two triangular transfer functions with identical widths. TCPMA data were collected for a wide range of CPMA resolutions, flow rates, and mass set points ranging from 2 to 15, 0.3 to 8 LPM, and 0.05 to 100 fg, respectively. The mass set point agreement remained relatively constant over a range of CPMA mass set points and increased slightly with decreasing CPMA resolutions. Neglecting outliers, the average mass set point agreement was suggesting good reproducibility among the CPMAs. The width factor showed a functional dependence on the mass set point, resolution, and flow rate. It was observed that the CPMA transfer function was generally narrower ( ) than the idealized transfer function except at a low flow rate (0.3 LPM) and low mass set points ( fg), and the width factor approached unity at higher mass set points ( fg) and higher resolutions ( ). As expected, the height factor depends on the mass set point, resolution, and flow rate: it decreases with lower mass set points, lower flow rates and higher resolutions. Both the width and height factors were fitted robustly using multivariate non-linear fitting models so that CPMA users can easily calculate its transfer function over a wide range of operating conditions. Copyright © 2023 American Association for Aerosol Research


Introduction
Developed by Olfert and Collings (2005), the centrifugal particle mass analyzer (CPMA) is an instrument that classifies particles by their mass-to-charge ratio. The CPMA is made up of two cylindrical electrodes rotating coaxially with a voltage potential between them. The cylinders, rotating at different speeds, impose simultaneous centrifugal and electrical forces on charged particles. Aerosols, depending on their mass and electric charge, experience varying centrifugal and electrical forces. The centrifugal force acts toward the outer cylinder, while the electrical force acts toward the inner cylinder for particles with the same polarity as the applied electric field. Conversely, for particles with opposite polarity, the electrical force acts toward the outer cylinder. As a result, larger aerosols are subjected to a larger centrifugal force, pushing them toward the outer cylinder, while smaller aerosols with the same polarity as the applied electric field are attracted toward the inner cylinder. Particle will exit the classifier if a balance between the electrical and centrifugal forces occurs. Sakai and Rothamer 2017;Graves, Koch, and Olfert 2017;Ubogu et al. 2018) and the measurement of particulates emitted from other sources, e.g., cigarette smoke (Johnson et al. 2014(Johnson et al. , 2015b(Johnson et al. , 2015a, gas flaring Sipkens et al. 2021), spark-discharge generators (Nilsson et al. 2015), plasma generator (Graves et al. 2020), and powdered mineral dust (Marsden et al. 2018). Moreover, pairing CPMAs with other aerosol classifiers in a tandem arrangement has yielded extensive information about such particles. In these tandem measurements, the CPMA is operated in various static modes, i.e., at a fixed mass set point, to allow aerosols of a particular mass-tocharge ratio to pass through the CPMA classification region and be measured by instruments downstream. Of particular note are measurements of the mass-mobility relationships of aerosols using a CPMA in tandem with a differential mobility analyzer (DMA; Knutson and Whitby (1975); Liu and Pui (1974)) and a condensation particle counter (CPC; Agarwal and Sem (1980)), (Sipkens et al. 2021, Sipkens, Olfert, and Rogak 2020a, Sipkens, Olfert, and Rogak 2020bAfroughi et al. 2019;Graves, Koch, and Olfert 2017;Quiros et al. 2015;Johnson et al. 2014;Ghazi et al. 2013;Olfert, Symonds, and Collings 2007). The pairing of a DMA and CPC is also known as a scanning mobility particle sizer (SMPS) if the setpoint of the DMA is continuously scanned. The mass-mobility measurements have typically been used to determine either summary parameters or the relationship between the distribution of desired parameters of measured aerosols, such as the effective density Johnson et al. 2014;Olfert, Symonds, and Collings 2007), dynamic shape factor (Beranek, Imre, and Zelenyuk 2012;Kuwata and Kondo 2009), and mass-mobility exponent (Shapiro et al. 2012). Using a similar methodology, some studies have paired the CPMA with other aerosol instruments, for instance, optical instruments (e.g., single-particle soot photometer; Liu et al. (2017); Sedlacek III et al. (2018); Broda et al. (2018); Naseri et al. (2021aNaseri et al. ( , 2021b; Sipkens et al. (2021) and cavity attenuated phase shift spectrometers (CAPS) (Dastanpour et al. 2017), or aerosol mass spectrometers (Bell et al. 2017;Marsden et al. 2018), to measure the average coating mass and optical properties, or chemical compositions of measured aerosols, respectively.
While the CPMA classifies particles by mass-tocharge ratio, the classified particles are not strictly monodisperse in mass because, in addition to singlycharged particles of the desired mass, doubly-charged particles of twice the desired mass (and particles up to the maximum number of charges in the aerosol) also pass through the classification region. When the CPMA operates at low rotational speeds (e.g., when it classifies relatively large aerosols), small uncharged particles may also pass through the classification region. Also, aerosols with mass-to-charge ratios marginally higher and lower than the selected setpoint also pass through the classifier due to the width of its transfer function. The CPMA transfer function is a probability distribution that describes the CPMA performance in the classification of particles with a chosen mass-to-charge ratio. The transfer function has profound implications for interpreting the particle distributions classified by the CPMA. Of note, in tandem measurement systems where the CPMA is used (e.g., the CPMA-SP2 or CPMA-SMPS), the CPMA transfer function plays a crucial role in evaluating the kernel function (i.e., the theoretical response) of the system, which is used for retrieving desired information via inversion (Rawat et al. 2016;Buckley et al. 2017;Broda et al. 2018;Rogak 2020a, Sipkens et al. 2021;Naseri et al. 2021aNaseri et al. , 2021b. Accordingly, it is essential to characterize the CPMA transfer function carefully. The theoretical CPMA transfer function was initially derived by Olfert and Collings (2005), using a finite difference scheme and mathematical methods. Sipkens, Olfert, and Rogak (2020c) revisited the evaluation of the CPMA transfer function, implementing novel numerical approaches to improve the computational efficiency of calculating the transfer function. Despite the theoretical ability to calculate the transfer function of the CPMA, to date, limited data have been collected regarding how the actual transfer function differs from the theoretical models. Olfert et al. (2006) fitted a CPMA transfer function based on theory to experimental data using PSL spheres and a prototype CPMA; however, the study only reported the agreement between the PSL mass and the apparent mass measured by the CPMA. The study did not measure how the width or height of the transfer function deviated from theory. Furthermore, no data are available for the commercially available CPMA (Cambustion Ltd., Cambridge, UK).
The transfer functions of the differential mobility analyzer (DMA; Knutson and Whitby (1975)) and aerodynamic aerosol classifier (AAC; Tavakoli and Olfert (2013)) have been experimentally characterized using a tandem measurement system Fissan et al. 1996;Birmili et al. 1997;Stratmann et al. 1997;Martinsson, Karlsson, and Frank 2001;Karlsson and Martinsson 2003;Collins et al. 2004;Li, Li, and Chen 2006;Johnson et al. 2018). In this system, two identical aerosol classifiers are employed in series (i.e., a tandem arrangement). The first instrument is configured so that it operates at one set point and the total number concentration of particles exiting the classifier is counted. The second instrument, downstream of the first, is then stepped or scanned over the range of particle sizes exiting the first classifier, and the number concentration of particles exiting the second classifier is measured as a function of its setpoint. Consequently, there is a distribution of particle counts measured at the exit of the second classifier as a result of the convolution of the transfer functions of the two classifiers. The actual transfer function or how it differs from an ideal case is estimated using data deconvolution or fitting routines (Li, Li, and Chen 2006), assuming that the two classifiers have transfer functions with identical widths.
In this study, the tandem CPMA methodology was used to characterize the CPMA transfer function. Following previous work on the DMA and AAC, it is assumed that the ideal transfer function of the CPMA is triangular and experiments are used to determine factors which account for deviations from the idealized assumption: namely the width factor (l), which accounts for diffusional broadening or other factors that may affect the width of the transfer function, and the height factor (g), which accounts for losses in the CPMA due to impaction or diffusion losses, etc. This study is of considerable significance as the experimental characterization of the CPMA transfer function can significantly enhance the analysis and inversion of data collected using a CPMA with or without other aerosol classifiers.

Theory and methodology
2.1. CPMA transfer functions Figure 1 shows a simple schematic of the Couette CPMA, in which aerosols flow between two cylinders rotating coaxially with a voltage applied between them.
Particles with different masses (m) and charge states (n q ) are exposed to centrifugal (F C ) and electrical (F E ) forces, such that a limited range of particles can pass through the classification region. The performance of the CPMA in classifying particles is characterized by a probability distribution known as a transfer function. A number of theoretical models have been developed for the Couette CPMA (Olfert and Collings 2005;Sipkens, Olfert, and Rogak 2020c) in order to numerically estimate the CPMA transfer functions. In contrast to the previous theoretical models of the CPMA transfer function (Olfert and Collings 2005;Sipkens, Olfert, and Rogak 2020c), which are computationally intensive, there is a simplified representation of the CPMA transfer function in the form of a triangular shape that has been implemented in the CPMA software (Cambustion Ltd., Cambridge, UK).
To derive the triangular transfer function, it is necessary to examine the motion of particles in the classification region (see Figure 1) in order to determine the mass range of particles which can pass through the classifier. Accordingly, the following assumptions will be made to simplify the equation of motion: (i) it is assumed that the channel is sufficiently narrow that the forces on the particles in the gap are constant with respect to radius and evaluated at the center of the gap, (ii) particle diffusion and image forces are negligible, (iii) particles acquire their terminal velocity instantly, 1 (iv) the flow field is quasisteady and the axial flow (v x ) in the classification region is plug flow, and (v) the gap between the coaxial cylinders is much smaller than the CPMA radii (D gap ( r). Consequently, the equation of motion of particles in the CPMA classification region in the radial (r) direction, can be expressed as where F D is the viscous drag force, and v r is the particle velocity in the radial direction. Using Stokes' law, the viscous drag force is expressed as: where d m is the mobility equivalent diameter of the particle, C c is the Cunningham slip correction factor, l f is the viscosity of the carrier gas, and B is the particle mechanical mobility. The electrical and centrifugal forces on an aerosol at a distance r from the rotational axis are respectively: and Here, r 1 and r 2 are the inner and outer cylinder radius, respectively, V is the potential difference applied between the inner and outer cylinders, and n q is the number of elementary charges on the aerosol particle, e is elementary charge ($1.6 Â 10 -19 C), and 1 This assumption holds true since the relaxation time of the particles is much shorter than their residence time in the classification region.
x is rotational speed of the aerosol. Since the forces on the particles are assumed to be constant across the narrow gap between cylinders, the centrifugal and electric forces are calculated at the center of the gap (x ¼ x c and r ¼ r c ). At a given operational condition of the CPMA (i.e., a particular rotational speed and a voltage applied between cylinders), the centrifugal and electrical forces become equal for particles with a certain mass and charge state, such that they transit axially along the classifier. As a result, at this equilibrium (i.e., F D ¼ 0) at the center of the classifier (i.e., r ¼ r c ), Equation (1) simplifies to F E ¼ F C , which expands to the following based on Equations (3) and (4): Rearranging Equation (5) in terms of the mass-tocharge ratio yields: c r 2 c ln r 2 =r 1 ð Þ : Throughout this paper, the CPMA setpoint reported only reflects singly-charged particles (i.e., n q ¼ 1) and is reported in fg. The particle with the greatest mass that can pass through the CPMA is the particle that enters the classification region at the inner edge of the classifier (point A in Figure 1b) and reaches the outer edge of the classifier at the end of the classification region (point B in Figure 1b).
Accordingly, Equation (1) for the maximum particle mass (m max, n q ) that exits the classifier with a charge of n q , is, where B max, n q is the mobility of the particle with m max mass and n q charge. The term on the right-hand side is zero as the particles reach their terminal velocity instantly 2 . Due to the plug flow assumption, the maximum radial particle velocity (v r, max ), which moves a particle radially from point A to B during its residence time in the classifier, is approximated by: where Q is the volumetric aerosol flow rate, D gap and L are the gap and the length of the classifier, respectively, and A D is the flow cross-sectional area. Substituting Equations (6) and (8) (with A D approximated by 2prD gap ) into Equation (7) gives, where the second term on the right-hand side is the full width at half maximum (FWHM) of the triangular transfer function in the mass domain. Consequently, the resolution of the triangular transfer function of the CPMA for particles with n q elemental charges, which is the dimensionless form of FWHM (normalized by the equilibrium mass), is estimated by, The voltage and rotational speed in Equation (5) are adjustable variables to obtain the user-selected mass set point (i.e., the equilibrium mass, m Ã ) and resolution. The supplementary information contains tables with the geometry of the CPMA and the operation conditions (rotational speed and voltage) for each measurement point (mass set point, resolution, and flow rate).
Alternative transfer functions have been derived. Olfert (2005) and Sipkens, Olfert, and Rogak (2020c) derived an identical expression (albeit calculated differently) for the CPMA transfer function by using the same assumptions as the triangular transfer function, except that they assumed the axial flow to be parabolic and forces to vary radially. Due to the variations in forces and the assumption of parabolic flow, the CPMA transfer function has a trapezoidal shape in their model. Furthermore, these studies presented an expression for the transfer function that incorporates particle diffusion, which here is referred to as the diffusion model. Figure 2 a illustrates the transfer function of the triangular model together with the trapezoidal and diffusion models of Sipkens, Olfert, and Rogak (2020c) (i.e., using Cases 1 C and 1 C-diff in that work) evaluated for the mass set point of m Ã ¼ 0.05 fg ($ 42 nm mobility diameter) with a constant flow rate of Q ¼ 0:3 LPM and resolutions of R m ¼ 2 and R m ¼ 15, respectively. It is observed that while the triangular model is symmetric around the equilibrium mass-to-charge ratio, the trapezoidal model is positively skewed (i.e., concentrated on higher particle masses) and has a narrower width at low resolution. The difference arises from the fact that while the forces are assumed to be constant in evaluation of the triangular model, they vary slightly with radius, such that the electrostatic and centrifugal forces are larger near the inner cylinder. As a result, smaller particles are removed more efficiently than larger ones, resulting in a slight skewness toward higher masses. Moreover, it can be seen that the diffusion model is more skewed than the trapezoidal model when the sample flow rate and resolution are low. This additional skewness is due to particle diffusion becoming greater as particle size decreases resulting in greater losses of smaller particles. A comparison between Figure 2 a and b reveals that as the resolution increases from a low resolution of R m ¼ 2 to a high resolution of R m ¼ 15, both trapezoidal and diffusion models become nearly symmetrical, and the width of trapezoidal and diffusion models approach that of the triangular transfer function.

Determining deviations from the triangular transfer function
The Results (Section 4) will show that none of the transfer function models accurately predict the actual transfer function of the CPMA. Therefore, an empirical model can be used to correct a theoretical model of the CPMA's transfer function. Theoretically, this could be done by applying correction factors to any of the idealized transfer functions described above. However, the diffusion and trapezoidal transfer functions are computationally expensive which prohibits their use on the hardware of the commercial CPMA, and they also make two-dimensional inversion very time-consuming. Thus, the work presented here will examine how the actual transfer function differs from the triangular transfer function and use correction factors to minimize the difference between them. An ideal triangular (i.e., non-diffusing) transfer function of the CPMA (X ideal ) in the domain of nondimensional particle mass,m ¼ m=m Ã , can simply be expressed as where b is the inverse of the CPMA resolution (1=R m ) and is equivalent to the FWHM of the triangular transfer function. The area of the prescribed transfer function (A) determines the number of particles passing through the classifier and is often referred to as the transmission efficiency. It is calculated by where h ¼ 1 is the height of the idealized triangular transfer function at the instrument set point. This simple expression is equivalent to the CPMA transfer function derived in Equations (9) and (10), but expressed in the domain of non-dimensional mass.
As a result of non-idealities in particle classification, the actual shape of the CPMA transfer function will differ from a triangle and may narrow or broaden the width, or increase or decrease the height of the transfer function relative to the idealized triangle. Previous, researchers using the DMA or AAC (Birmili et al. 1997;Martinsson, Karlsson, and Frank 2001;Johnson et al. 2018) have used various adjustable factors to correct Equation (11) to account for non-idealities, although the definitions of the adjustable factors vary between studies. Following the work of Birmili et al. (1997), two adjustable factors are introduced to Equation (11) as where g and l are height factor and width factor, respectively. While the height factor scales the transfer function's integrated area (A) by modifying its height (h) to account for particle losses (e.g., losses occurring through diffusion and impaction), the width factor adjusts the transfer function's width (FWHM nonÀideal ¼ b=l) to account for its potential broadening due to particle diffusion or any other non-idealities in the CPMA (e.g., flow effects). An illustration of the idealized transfer function (Equation (11)) and how its width (FWHM), height (h), and area (A) are modified by the adjustable factors (i.e., non-idealized transfer function, Equation (12)) are shown in Figure 2c. In this example, the non-idealized transfer deviates from the idealized transfer function with l and g < 1, demonstrating that along with losses, the actual transfer function may be broader than the idealized one. The adjustable factors g and l are derived by conducting the tandem CPMA measurements as shown in Figure 3a. In the experiment, the number concentration of aerosols downstream (N 2 ) and upstream (N 1 ) of the second CPMA are measured with a CPC as the second CPMA is stepped through a range of set points. The theoretical number concentrations can also be calculated through the convolution of the incoming particle count distribution and the CPMA transfer functions, where dN=dlogm i is the number distribution of aerosol entering the first CPMA, g CPC ðm Ã Þ is the counting efficiency of the CPC at a mass set point (m Ã ), X is the non-idealized triangular CPMA transfer function, in which the subscript 1 and 2 refer to the first and second CPMA, and m Ã 12 represents the mass set point agreement between the two CPMA set points (i.e., m Ã 12 ¼ m Ã 2 =m Ã 1 ). This factor was included in Equation (13) to account for potential discrepancies between  (11)) and non-idealized (Equation (12)) triangular CPMA transfer functions.
the mass set points of the two CPMAs due to minor differences between the classifiers (e.g., differences in voltage and rotational speed calibrations, or classifier dimensions). The equation is simplified by assuming that the counting efficiency of the CPC is constant over the narrow range of particle masses stepped through by the downstream CPMA, i.e., g CPC ðm Ã 1 Þ ¼ g CPC ðm Ã 2 Þ: It is also assumed that width factors of both CPMA transfer functions were identical (l 1 ¼ l 2 ). (Note, the relative resolution of both CPMAs (R m or b) is the same at all set points in an experiment; however, the absolute width of the transfer function of the second CPMA (e.g., m max À m Ã ) changes slightly at each set point). Additionally, as N 1 was directly measured, all losses within the upstream CPMA were already taken into account, and accordingly, g 1 ¼ 1 was used.
Minimizing the difference between the theoretical and experimental values of the N 2 =N 1 concentration ratio (Equation (13)) using chi-squared minimization determined g, l and m Ã 12 : This analysis assumed a low likelihood of multiply-charged aerosols at the entrance of the first CPMA so that their effects on the convolution would likewise be negligible. It is shown in the Section S2 of Supplementary material that the fraction of multiply-charged aerosols was very small and does not have a significant affect on the results.
Unlike previous studies with DMAs and AACs, which assumed a uniform number concentration of aerosol over the narrow width of the upstream classifier, (e.g., Martinsson, Karlsson, and Frank (2001) and Johnson et al. (2018)) the number distribution of aerosol particles upstream of the first CPMA was measured by an SMPS and considered in the convolution. This additional consideration was largely needed to account for the effects of using an AAC to remove small particles prior to TCPMA classification (as discussed in the next section) and helped increase the accuracy of the TCPMA convolution.

Tandem CPMA measurements
A schematic of the TCPMA experimental setup is shown in Figure 3a. Since the width of the CPMA transfer function and its calculation depends on particle mobility, it was necessary to measure an aerosol with known mobility. To this end, a $ 1% solution of Santovac 5 in acetone (purity ! 99.5%) was used to generate spherical aerosol particles with a known density (1198 kg/m 3 ). The TSI atomizer (Model 3076) was employed to atomize the solution and generate the aerosol. In order to evaporate the acetone, the generated aerosol was diluted by a factor of $ 25 with dilution air at a temperature of 60 C using a twostage diluter (Dekati V R eDiluter Pro). TCPMA measurements can become complicated when small uncharged aerosols potentially pass through the classifier at very low rotational speeds/low centrifugal forces (e.g., when the CPMA operates at high mass set points and low resolutions). Consequently, an AAC (Cambustion Ltd., Cambridge, UK) was used upstream of the tandem CPMAs to remove small particles that were outside the CPMA transfer function which could have passed through the CPMA if they were uncharged. The AAC was set to operate at low resolutions (e.g., R s ranging between 1.5 and 4), so that a wide range of particle sizes could pass through it and enter the first CPMA. Furthermore, the AAC aerodynamic diameter set points were selected so that the mode of AAC-classified distributions would always be slightly smaller than the mass equivalent diameters of the first CPMA mass set points. This AAC set point selection was undertaken to maximize the number of aerosols entering the TCPMA system while minimizing the portion of multiply-charged particles larger than that of singly-charged ones. The mobility size distributions of AAC-classified aerosol particles were then measured by a scanning mobility particle sizer (SMPS, TSI Inc.) for the TCPMA data convolution, as previously explained in Section 2.
In the measurements taken at flow rates of 0.3 or 1.5 LPM, the TCPMA aerosol flow rate was controlled by an internal flow controller within the condensation particle counter (CPC; TSI, Model 3775). For the measurements with higher flow rates (i.e., 4 and 8 LPM), the CPC was operated at a flow rate of 1.5 LPM, and a vacuum pump and a mass flow controller (MFC; Alicat Scientific, MCV-Series: Vacuum Gas Mass Flow Controllers, serial number 175509) were employed together to extract an additional flow of 2.5 or 6.5 LPM, resulting in a net flow of 4 or 8 LPM through the TCPMA system, respectively. Additionally, because the AAC cannot operate at flow rates higher than 3 LPM, the classified aerosol flow from the AAC was supplemented with filtered compressed air controlled with a mass flow controller (MFC; Alicat Scientific, MC-series, serial number 270591) to maintain the desired flows. A static mixer (Ko-flo Corporation, model number 3/8-40-3-12-2) was used to generate a well-mixed aerosol flow with the added filtered air prior to measurement by the TCPMA system.
Prior to TCPMA measurement, the aerodynamically classified aerosol particles were electrically charged by passing them through a radioactive Kr-85 bipolar diffusion charger (TSI Inc., Model 3077 A). The first CPMA (CPMA Mark I, Cambustion Ltd., serial number C220) was operated at a fixed mass set point of m Ã 1 ¼ 0.05, 0.1, 1, 10, or 100 fg, while the downstream CPMA (CPMA Mark I, Cambustion Ltd., serial number C313) was stepped (m Ã 2 ) through the range of classified particles, and the number concentration of the twice-classified particles (N 2 ) was measured with the CPC. At the beginning and end of each TCPMA measurement, the second CPMA was bypassed to measure the steady-state number concentration of aerosol particles passing through the upstream CPMA (N 1 ) during the measurement.
The CPMAs were examined at five mass set points between 0.05 -100 fg, four flow rates ranging from 0.3 -8 LPM, and five resolutions (R m ) of 2, 3, 6, 10 or 15 if the operating conditions fell within the CPMA's operating window (i.e., it was not possible to test certain conditions because the voltage or rotational speed of the CPMA would have exceeded the maximum or minimum). There were some measurements that were within the operating envelope of the CPMAs; however, the experiments could not be completed due to high motor temperature errors. Each TCPMA configuration was repeated three times in order to assess the measurement repeatability. Also, a series of measurements, covering mass set points ranging from 0.05 fg to 100 fg, with a flow rate of 1.5 LPM and CPMA resolution of 6, were repeated with the positions of the two CPMAs swapped. This was done to gain insights into the differences in their transfer functions, specifically the mass set point agreement and height factor, as discussed in Section 4.1 and the supplementary information. The supplementary information also contains a complete list of all data collected in the study.

Stationary loss measurements
In addition to the CPMA losses characterized as height factor from the TCPMA measurements, the particle losses were measured when the CPMA was not operating (i.e., no rotational speed and no voltage). Figure 3b shows a schematic of the stationary loss measurement setup. In this measurement, the SMPS was used to measure the number concentration as a function of particle mobility diameter upstream (N 1 ) and downstream (N 2 ) of the CPMA when it was not operating. These measurements approximate the TCPMA case of R m ¼ 0, which were made at flow rates of 0.3, 1.5, and 4 LPM. Figure 4 depicts the concentration ratios (N 2 =N 1 ) derived from the TCPMA measurements (i.e., convolution of the two CPMA transfer functions) conducted at a flow rate of 0.3 LPM for five CPMA mass set points ranging from 0.05 to 100 fg, with a resolution of R m ¼ 6, and compares them with their theoretical counterparts, which are the convolution of two theoretical transfer functions: (i) triangular model, (ii) trapezoidal, and (iii) diffusion model, shown as dashed green, solid red, and blue lines, respectively. Compared to their ideal counterparts, the observed N 2 =N 1 ratios generally had lower amplitudes and a slight offset, and they were either slightly wider for low mass set points, e.g., m Ã ¼ 0.05 and 0.1 fg, or narrower for high mass set points. It can be seen that the convolution for each of the three theoretical models could not accurately represent the measured N 2 =N 1 : In contrast, the convolution of two non-idealized CPMA triangular functions is able to fit the data well (solid black lines), indicating that a reasonable approximation of the actual CPMA transfer function had been obtained by using the triangular transfer function with three adjustable factors (g, l and m Ã 12 ). The following sections describe the results of the three factors for the rest of the data set. Figure 5a illustrates the mass set point agreement (m Ã 12 ) between the two CPMAs, such that a value of 1 indicates that the mass set points of the two CPMA are equivalent. The potential discrepancies between the mass set points of the two CPMAs are due to minor differences between the classifiers (e.g., differences in voltage and rotational speed calibrations, or classifier dimensions). It is observed that the agreement has a small dependence on mass set point and resolution, as reflected in Figure 5a, such that higher CPMA resolutions (reflected by series color in the figure) tend to result in a closer agreement between the CPMA's mass set points at larger mass set points. In the current study, the measurements with mass set point agreements greater than 1.15 or lower than 0.85, have been regarded as outliers as they do not conform to the main assumption underlying the TCPMA measurements (running two identical CPMAs in series). The outliers are identified by data points outlined in red in Figure 5a, which are associated with low mass set point measurements conducted at a low flow rate of Q ¼ 0:3 LPM and low resolutions of R m ¼ 2 and 3. The presence of outliers in this study is likely attributed to two issues. Firstly, the combination of low mass set points, low flows and low resolutions results in low classifier voltages (< 4 V) which are difficult to accurately control. Our independent measurement of voltage with an external voltmeter revealed that voltage errors of up to 10% were found on one CPMA below 10 V, while the errors were less than 2% above 10 V. Secondly, determining the m Ã 12 is inherently less accurate when the measured distribution of (13)) is broad due to low-resolution classifiers (i.e., peak-finding is less accurate on broad distributions). As shown in Figure 5a the low resolution set points have more scatter at all set points compared to higher resolution set points. Neglecting the outliers, the average mass set point agreement between the CPMAs is m Ã 12 ¼ 1:0260:03, indicating good reproducibility among the CPMAs. This outcome was further evaluated by swapping the positions of the CPMAs as shown in Figure 5b, which shows the mass set point agreements of the original CPMA arrangement and when the CPMA position is swapped for mass set points ranging between 0.05 fg to 100 fg, with Q ¼ 1:5 LPM and R m ¼ 6: The nature of the experiment calls for symmetry around 1 (here with an average error of $1.5 %), such that when CPMA positions are swapped, the magnitude of the offset is expected to remain the same while shifting in opposite direction (inversed). These results imply that swapping the CPMA positions does not have a major effect on the results. The measured width factors are represented as discrete points and the CPMA resolution is indicated through color gradients. The dot-dashed and dotted lines represent the theoretical width factor for the trapezoidal and diffusion transfer functions, respectively, if they were the actual transfer function. To derive the theoretical width factor, synthetic data of N 2 =N 1 was generated using Equation (13) using either the trapezoidal or diffusion transfer function to calculate X 1 and X 2 : It was assumed that the particle number distribution entering the first CPMA (dN=dlogm) is uniform over the width of the CPMA transfer function (i.e., not dependent on particle mass), and the counting efficiency of particles (g CPC ðm Ã Þ) and the mass set point agreement between the two CPMA set points (m 12 ¼ m Ã 2 =m Ã 1 ) were one for all mass set points. The synthetic data was then subjected to deconvolution using the triangular model, following the same procedure described previously for the experimental data. The synthetic data was calculated for a wide range of mass (m 1 Ã ) values, from 10 À4 fg to 200 fg.

Width factor (l)
The theoretical width factor for the trapezoidal model predicts that the CPMA resolution is the main factor affecting the width factor for all flow rates, with a minimal impact from the mass set point in highflow measurements. This outcome is supported by the TCPMA data plotted concurrently. Referring back to Figure 2a, the actual transfer function, which is likely better represented by the trapezoidal or diffusion model, is expected to be narrower than the triangular transfer function (except for a flow rate of 0.3 LPM, where diffusion results in a broader transfer function). As such, the convolution of two narrower trapezoidal transfer functions, yields width factors greater than one, which decrease as the resolution increases and approach unity at higher resolutions. The weak dependence of the width factor on the mass set point is because the width of a CPMA transfer function is largely determined by the particle electrical mobility and its relative change with particle mass along the range of mass set points. The diffusion model, which gives the identical width factor as the trapezoidal model for larger particles (m Ã ) 1 fg), predicts that diffusion decreases the width factor for smaller particles as expected-resulting in a broader transfer function than the ideal triangle model. Diffusion effects are a function of flow rate and are expected to become more apparent as flow rate decreases at lower mass set points as the smaller particles will have a longer residence time in the classifier and can diffuse more efficiently.
The measured data confirms the predictions of the theoretical trapezoidal model, indicating that the CPMA resolution is the primary factor affecting the width factor. The impact of particle mass on width factors is only observed at low flow rates (0.3 LPM) and low mass set points, as shown in Figure 6a. It should be noted that the absence of significant diffusion effects is due to the fact that the particle size studied experimentally was not small enough to observe this phenomenon. The width factor also demonstrates a small dependence on particle mass, as expected.
These findings were verified by a three-way ANOVA analysis, which quantitatively evaluated the significance of the CPMA settings (m Ã , R m , Q) on the Figure 6. The width factor derived from TCPMA measurements and synthetic data generated from the convolution of the trapezoidal and diffusion models. The measured and the corresponding theoretical width factor derived for different flow rates (Q) ranging from 0.3 to 8 LPM, at five CPMA mass set points (m Ã ) between 0.05 and 100 fg, with five different resolutions (R m ) ranging from 2 to 15. width factor. The results, sorted by p-value for three groups of data: all measurements, low flow measurements (Q ¼ 0:3 LPM), and high flow measurements (Q > 0:3 LPM), are listed in Table 1. The p-value indicates the significance of a parameter's impact on an adjustable factor; a smaller p-value means a more significant effect. The ANOVA analysis performed on all flow rates (first column of Table 1) showed that Q was the most significant input parameter in explaining the variations in the width factor, with a p-value of $ 10 À16 : The analysis using only low-flow data (second column) revealed that the mass set point (m Ã ) was only significant at low flow rates. The analysis using only data at high flow rates (third column) confirmed that the CPMA resolution (R m ) had a significant impact on the width factor, while Q and m Ã had negligible impacts.
Although the trends in the experimental observations align with theoretical predictions, the absolute values of the theoretical width factors differ from the experimental observations. This suggests that other mechanisms inside and outside the classification region may be important but are not accounted for in the transfer function models. For example, the models neglect entrance effects and assume fully developed, laminar flow at the entrance of the classification region, while in reality, there is an entry length to establish fully-developed flow. Additionally, the models assume an even distribution of particles across the classifier gap, but flow effects may result in non-uniform concentration at the entrance. Thus, since current theoretical models of the transfer function cannot accurately model the resolution of the CPMA, an empirical model is needed.
An empirical model was found using multivariate non-linear curve-fitting of the width factor data. An iteratively reweighted least squares (Dumouchel et al. 1989;Holland and Welsch 1977) routine was implemented in MatlabV R with the nlinfit function using the Welsch weight function. From this fitting procedure, the following equation was derived The length of the solid lines represent the operating region of Cambustion's commercial CPMA, such that the CPMA cannot be operated above or below the mass set point at the flow rate and resolution indicated. It is worth mentioning that although it is theoretically possible to run the CPMA for particle masses smaller than 0.05 fg at a flow of 0.3 LPM, in reality, the particle losses are so high that reliable data could not be collected. 2 Figure 7d shows the distribution of measured versus predicted values of width factor along with a 95% prediction interval, which is defined as an interval which contains 95% of the data (60:105). The parity plot shows that the majority of points lie near the diagonal, which is a positive indication that the fit model was acceptable. The determined prediction interval was added to Equation (14) as an upper and lower bound to illustrate the accuracy of the fit. For reference, the root-mean-square error for the fit is 0.0651. Figure 8a-c illustrate the height factor derived from TCPMA measurements as a function of mass set point Table 1. Three-way ANOVA table indicating the significance of the CPMA sampling configurations (i.e., m Ã , R m , Q) to the width and height factors. The analysis was conducted using the logarithm of mass set points, and all parameters were treated as continuous. As shown later in Figure 8 a, the stationary height factor (i.e., transmission efficiency) is $ 0.3 for particles at $0.05 fg in a single CPMA at Q ¼0.3 LPM. These particle losses would then be approximately doubled when the two CPMAs are operated in the tandem arrangement.

Height factor (g)
at flow rates of 0.3, 1.5, 4 and 8 LPM, respectively. The height factor is a measure of the fractional loss of particles within the CPMA with the ideal transfer function having a value of one and a lower value indicating lower transmission efficiency (in either case when the width factor is concurrently one). In addition to the height factors derived from the TCPMA measurements, the particle losses were measured when the CPMA was not operating (no rotational speed and no voltage) using SMPS measurements upstream and downstream of the stationary CPMA. This is an approximation of the case of R m ¼ 0, and the measurements were made with flow rates of 0.3, 1.5, and 4 LPM. These SMPS measurements provides insights into the CPMA's performance at small particle masses where TCPMA measurements were not possible. The CPMA resolution is reflected by the color gradations in Figure 8a-c to illustrate its effects on height factor. A distinct slope in Figure 8a-c indicates that height factors are highly dependent on particle mass, such that the height factor  (14)) to distribution of measured width factor are illustrated as solid lines at a corresponding flow rate and CPMA resolution. (d) Parity plot of the width factor correlation (Equation (14)) based on the TCPMA measurements.
decreased with the CPMA mass set point. This trend is supported by the ANOVA findings shown in Table 1 (fourth column), which shows that a variation of the height factor was predominantly determined by the mass set point of the CPMA. Also, ANOVA suggests that the CPMA resolution also contributed to the variation of the height factor to a considerable extent. This functional dependence on CPMA resolution can be seen as a vertical layer of color in Figure 8a-c, such that higher resolutions (moving from light to dark color) result in a lower height factor. Additionally, as can be seen from the ANOVA results in Table 1, the p-value for flow rate is close to 0.05, implying that the loss factor did not functionally vary with CPMA flow rate for the range of set points tested however this data set did not include the stationary measurements at mass set points lower than 0.05 fg, which clearly show a flow rate dependence.  (15)) to distribution of measured height factor are illustrated as solid lines at a corresponding flow rate and CPMA resolution. (d) Parity plot of the height factor correlation (Equation (15)) based on the TCPMA measurements.
Part of the losses in the CPMA can be attributed to inertial impaction, which is caused by the centrifugal forces acting on the particles outside of the CPMA's classification region. At higher CPMA resolution, the rotational speed of the cylinders for the same mass set point is higher, so higher impaction losses are expected for particles moving through the turns before and after the classification region. Impaction, however, cannot account for the lower height factor of lower mass set points at a given resolution. At smaller particle masses, losses are caused by diffusion which maybe be enhanced by turbulent flow outside of the classification region. Figure 8a-c also depict the height factor derived from the synthetic data fabricated by convolution of theoretical trapezoidal and diffusion transfer functions for flow rates of Q ¼ 0.3, 1.5, 4 and 8 LPM, at CPMA mass set points ranging from 0.0001 to 100 fg, with five different resolutions ranging in R m from 2 to 15. The theoretical height factors are greater than one (except where particle diffusion dominates, e.g., at low flow rate and low mass set points), which is caused by the larger area of the theoretical models compared to the idealized triangular transfer function. It is evident that theoretical models do not correspond to experimental observations. The substantial discrepancy between the theoretical and measured height factors implies that there might be some effects, both inside and outside of the classification region, which have been neglected while developing the models. In addition to the assumptions mentioned in the previous section, the theoretical models assume no forces act on particles outside of the classification regions. In contrast, particles are subjected to both centrifugal and electrostatic forces while moving through the turns before and after the classification region. Therefore, the current theoretical models of the transfer function cannot accurately describe the loss of the CPMA, and an empirical model is necessary.
Accordingly, the height factors derived from TCPMA measurements for each flow rate were fitted to a single form of multivariate nonlinear model. The stationary data were incorporated into the fit to constrain it at small particle masses where TCPMA measurements were impossible.
The fitted relation found was, where R m is the CPMA resolution, and m Ã is the CPMA mass set point in fg; respectively. Table 2 shows the required coefficients of Equation (15) for each flow rate. Figure 8a-c show the results of fitted model to distribution of measured height factor as solid lines at four different flow rates of 0.3 to 8 LPM. The length of solid lines represents the theoretical operating region of the CPMA. As can be seen in Figure 8, there is a very narrow operational range in which the CPMA can operate at Q ¼ 8 LPM, and the height factor remains constant throughout that range. Therefore, the regression coefficients in Equation (15) were set in such a way that the fit model gives a constant result. It is significant to note that this fit is only valid over the range of data collected, and the height factor for flow rates other than those measured in this study can be derived by interpolation. Figure 8d shows the distribution of measured versus predicted values of height factor based on the TCPMA measurements for all four flow rates along with a 95% prediction interval (60:143). The parity plot indicates that most points are near a diagonal, which is a positive indication that the fit model was acceptable. In the case of the fit, the root-mean-square error was found to be 0.0876.

Conclusions
This study examined the transfer function of the CPMA by using a tandem CPMA (TCPMA) configuration. Following the methodology described, a triangular transfer function was used to approximate the CPMA transfer function, and deviations from the idealized triangular transfer function were determined by three factors: (i) the mass set point agreement, (ii) the width factor, and (iii) the losses factor. The Table 2. The regression coefficients of the fitted expression for height factor at four flow rates of Q ¼ 0.3, 1.5, 4, and 8 LPM. measurements with mass set point agreements greater than 1.15 or lower than 0.85 were considered as outliers in this study as they do not conform to the main assumption underlying TCPMA measurements (i.e., running two identical CPMAs in series). Excluding the outliers, the average mass set point agreement between the CPMAs was m Ã 12 ¼ 1:0260:03, indicating good reproducibility among the CPMAs. The width factor was typically found to be higher than one (l > 1), indicating that the CPMA transfer function was narrower than the idealized triangular transfer function. Nevertheless, a broader CPMA transfer function (l < 1) was observed when the CPMA was run at low mass set points (i.e., m Ã < 1 fg) and low flow rates (Q ¼ 0:3 LPM). The actual CPMA transfer function significantly deviated from the idealized triangular transfer function in terms of the height factor, which was influenced by the mass set point, resolution, and flow rate. The height factor decreased with a decrease in the CPMA mass set point and flow rate, and with a increase in the resolution. Multivariate non-linear models were fitted to the TCPMA measured data in order to capture the functional variations in the CPMA width and height factors. The CPMA empirical transfer function, as defined by Equations (14) and (15), is available from Naseri (2023).
There are two limitations to this study. Firstly, it is expected that the height factor, and to a lesser extent, the width factor will be a function of the mass and the mobility of the particles. The relationship between the mass and mobility of a particle, which is a function of the particle morphology and material density, is often reported as the effective density. Differences in effective density will result in differences in diffusional and inertial losses, affecting the height factor. These losses are difficult to model outside of the classification region, and the changes in height factor due to changes in effective density could not be predicted. In this study, we specifically generated spherical particles with a material density of 1198 kg/m 3 (in this case the effective density is equal to the material density) in order to characterize the CPMA transfer function. This effective density is within a factor of two of the effective density of most aerosol particles (a notable exception being spherical metal particles). For small particles, changes in particle diffusion will affect the width factor. Theoretical investigation shows (in the supplementary information) that a wide range of effective densities (500 to 2000 kg/m 3 ) has little affect on the width factor in the practical operating range of the CPMA.
The second limitation is that only the Mark I CPMA was tested. Cambustion has recently released a Mark II version of the instrument, which uses the same classifier as the Mark I version but has a different inlet/outlet configuration. Since the classifier is the same, it is expected that the width factors of the two versions would be similar. However, since inlet/outlet losses were found to be significant, there may be differences in height factor between the two versions.