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\begin{document}

\begin{frontmatter}

\title{Supplementary Information\\ for\\ Measurement of the centrifugal particle mass analyzer
transfer function}

\author[1]{A. Naseri\,\orcidlink{0000-0002-1493-6761}}
\author[2]{Tyler J. Johnson\,\orcidlink{0000-0002-3641-0381}}
\author[3]{Greg Smallwood\,}
\author[1]{J. S. Olfert\corref{cor1}\,\orcidlink{0000-0002-9681-5428}}
\affiliation[1]{organization={Department of Mechanical Engineering, University of Alberta},%Department and Organization
            city={Edmonton}, 
            state={Alberta},
            country={Canada}}
\affiliation[2]{organization={Atmose Ltd},
            city={Edmonton},
            state={Alberta},
            country={Canada}}
\affiliation[3]{organization={Metrology Research Centre,National Research Council Canada},
Department and Organization
           city={Ottawa}, 
           state={Ontario},
           country={Canada}}
\end{frontmatter}



In the following sections of this supplementary information, we provide additional information regarding the geometric characteristics of the CMPAs (Mark I CPMA, Cambustion) used in the current study, as well as operational parameters including rotational speed ($\omega$) and voltage ($V$) settings used for controlling the selected mass setpoints and resolutions during the TCPMA measurements. Furthermore, we provide an overview of all the tandem CPMA measurements conducted, along with the fraction of positively charged particles predicted to exit the first CPMA in a multiply-charged state.


\section{Geometric characteristics of employed CPMAs}
\label{sec:GeometricCharacteristics}
The geometric characteristics of the CPMAs (Mark I CPMA, Cambustion) used in the current study are listed in Table S\ref{tab:tabE}.

\begin{table}[!htbp]
\centering
\caption{Dimensions correspond to a commercially-available CPMA (Mark 1, Cambustion) used in this study.\label{tab:tabE}}

\fontsize{4}{3}\selectfont
\resizebox{\columnwidth}{!}{%
\begin{tabular}{lccccc}
\hline\hline
\\
Property & $r_1$& $r_2$  & $L$ & $\Delta_\mathrm{gap}$& $\bar{\omega}$ \\ \\
\hline \\
Value & 60 mm& 61 mm& 200 mm& 1 mm& 0.9697 \\ \\
\hline
\end{tabular}
}
\end{table}

\section{Effects of multiple charges and summary of data}
\label{sec:multipleCharges}

This study evaluates the CPMA transfer function assuming that the aerosol particles are only singly charged; however, the charge neutralizer employed is capable of imparting multiple charges to the particles. Assuming the number of elemental charges ($n_\mathrm{q}$) varies from 1 to 5 on the aerosol particles, and as the standard commercial CPMA only classifies positively-charged particles, the fraction of positive multiply-charged particles that enter the second CPMA ($F_{n_\mathrm{q}=1}$) can be calculated as
\begin{equation}
\label{Eq:A1}
F_{n_\mathrm{q}>1}=1-F_{n_\mathrm{q}=1} \approx 1-\frac{\int \mathrm{d}N/\mathrm{dlog}m\, \Omega_{1}\left(m_{1}, m_{1}^{\ast}, \eta_{1}, \mu_{1}\right)f\left(n_\mathrm{q}=1, m_{n_\mathrm{q}=1}\right)d m}{\sum_{n_\mathrm{q}=1}^{5} \int \mathrm{d}N/\mathrm{dlog}m \, \Omega_{1}\left(m_{1}, m_{1}^{\ast}, \eta_{1}, \mu_{1}\right)f\left(n_\mathrm{q}, m_{n_\mathrm{q}}\right)d m},
\end{equation}
where, $\mathrm{d}N/\mathrm{dlog}m$ is the number distribution of aerosol entering the first CPMA, and $m_{n_\mathrm{q}}$ is the mass of particles having the equivalent mobility diameter of $d_{n_\mathrm{q}}$, and $f\left(z_{n_\mathrm{q}}, m_{n_\mathrm{q}}\right)$ represents the charge fraction of particles of a certain mass ($m_{n_\mathrm{q}}$) and charge state ($m_\mathrm{q}$). Traditionally, the charging function is calculated using \cite{gunn1956measurements} as 

\begin{equation}
    f\left(z_{k}, d\right)=\frac{1}{\left[2 \pi \sigma_{\mathrm{z}}(d)\right]^{1 / 2}} \exp \left\{-\frac{1}{2} \frac{\left[z_{k}-\sigma_{\mathrm{z}}(d) \ln \left(Z_{\mathrm{Z}}\right)\right]^{2}}{\sigma_{\mathrm{z}}(d)}\right\}
\end{equation}
where $Z_{\mathrm{z}}$ is the ion mobility ratio which is approximated by $ \sim 0.875$ \citep{wiedensohler1986investigation}. Also,
\begin{equation}
    \sigma_{\mathrm{Z}}(d)=\frac{2 \pi \varepsilon_{0} d k_{\mathrm{B}}T}{e^{2}},
\end{equation}
where $e$ is the elementary electron charge ($e=1.6\times10^{-19}$ C), $\varepsilon_{0}$ is the dielectric constant, $k_{\mathrm{B}}$ is the Boltzmann constant, and $T$ is the aerosol temperature. 

Table S\ref{tab:tableD} illustrates a summary of the results measured from the TCPMA experiments, along with the fraction of positive multiply-charged particles entering the measurement system. The fraction of positive multiply-charged aerosol particles at the entrance of the second CPMA is about $ 2\%$ on average, which has a negligible effect on the data analysis.

\begin{table}[htbp!]
\centering
\caption{An overview of all tandem CPMA measurements as well as the fraction of positive multiply-charged particles exiting the first CPMA. The measurements marked by $\dagger$ are regarded as outliers and not included in the fits. The data presented in the table are expressed as mean values accompanied by 95\% confidence intervals.\label{tab:tableD}}

\textcolor{black}{\fontsize{21}{13}\selectfont
\resizebox{\columnwidth}{!}{%
\begin{tabular}{lccccclclclccccccccccccccc}
\multicolumn{24}{c}{Outline of TCPMA data} \\ \\ \hline\hline \\
 &
  $Q$&
   &
  &
   &
  $m^*$ &
   &
  \multicolumn{1}{l}{$d^*$} &
   &
  $V$&
   &
  $\omega$ &
   &
  \multicolumn{3}{c}{$\mu$} &
   &
  \multicolumn{3}{c}{$\eta$} &
   &
  \multicolumn{3}{c}{$m^*_{12}$} &
   &
  $F_{n_\mathrm{q}>1}$ \\
&$_\mathrm{[LPM]}$&
   &
  $R_m$ &
   &
  $_\mathrm{[fg]}$ &
   &
  \multicolumn{1}{l}{$_\mathrm{[nm]}$} &
   &
  $_\mathrm{[V]}$&
   &
  $_\mathrm{[rad/s]}$&
   &
  \multicolumn{3}{c}{} &
   &
  \multicolumn{3}{c}{} &
   &
  \multicolumn{3}{c}{} &
   & \\
\cline{2-2} \cline{4-4} \cline{6-6} \cline{8-8} \cline{10-10} \cline{12-12} \cline{14-16} \cline{18-20} \cline{22-24} \cline{26-26} 
  \\
$\dagger$ &
  0.3 &
   &
  2 &
   &
  0.101 &
   &
  54.4 &
   &
  1.85 &
   &
  220 &
   &
  1.26 &
  $\pm$ &
  0.01 &
   &
  0.54 &
  $\pm$ &
  0.03 &
   &
  0.84 &
  $\pm$ &
  0.00 &
   &
  0.00 \\
 &
  0.3 &
   &
  2 &
   &
  1.03 &
   &
  117 &
   &
  7.29 &
   &
  137 &
   &
  1.42 &
  $\pm$ &
  0.02 &
   &
  0.96 &
  $\pm$ &
  0.02 &
   &
  0.93 &
  $\pm$ &
  0.02 &
   &
  0.02 \\
 &
  0.3 &
   &
  2 &
   &
  10.4 &
   &
  255 &
   &
  24.6 &
   &
  78.9 &
   &
  1.46 &
  $\pm$ &
  0.01 &
   &
  1.12 &
  $\pm$ &
  0.02 &
   &
  0.97 &
  $\pm$ &
  0.00 &
   &
  0.06 \\
$\dagger$ &
  0.3 &
   &
  3 &
   &
  0.051 &
   &
  43.2 &
   &
  1.66 &
   &
  295 &
   &
  0.96 &
  $\pm$ &
  0.01 &
   &
  0.32 &
  $\pm$ &
  0.01 &
   &
  1.25 &
  $\pm$ &
  0.06 &
   &
  0.00 \\
$\dagger$ &
  0.3 &
   &
  3 &
   &
  0.101 &
   &
  54.4 &
   &
  2.58 &
   &
  260 &
   &
  1.05 &
  $\pm$ &
  0.01 &
   &
  0.51 &
  $\pm$ &
  0.01 &
   &
  1.22 &
  $\pm$ &
  0.04 &
   &
  0.01 \\
 &
  0.3 &
   &
  3 &
   &
  1.01 &
   &
  117 &
   &
  10.2 &
   &
  163 &
   &
  1.20 &
  $\pm$ &
  0.01 &
   &
  0.84 &
  $\pm$ &
  0.05 &
   &
  1.00 &
  $\pm$ &
  0.01 &
   &
  0.02 \\
 &
  0.3 &
   &
  3 &
   &
  10 &
   &
  252 &
   &
  69.5 &
   &
  135 &
   &
  1.21 &
  $\pm$ &
  0.02 &
   &
  0.88 &
  $\pm$ &
  0.06 &
   &
  1.00 &
  $\pm$ &
  0.00 &
   &
  0.05 \\
 &
  0.3 &
   &
  3 &
   &
  96.9 &
   &
  537 &
   &
  87.3 &
   &
  48.8 &
   &
  1.24 &
  $\pm$ &
  0.01 &
   &
  1.07 &
  $\pm$ &
  0.02 &
   &
  0.99 &
  $\pm$ &
  0.00 &
   &
  0.13 \\
 &
  0.3 &
   &
  6 &
   &
  0.05 &
   &
  43.2 &
   &
  3.06 &
   &
  401 &
   &
  0.79 &
  $\pm$ &
  0.01 &
   &
  0.20 &
  $\pm$ &
  0.01 &
   &
  1.12 &
  $\pm$ &
  0.01 &
   &
  0.01 \\
 &
  0.3 &
   &
  6 &
   &
  0.102 &
   &
  54.5 &
   &
  4.77 &
   &
  352 &
   &
  0.90 &
  $\pm$ &
  0.00 &
   &
  0.32 &
  $\pm$ &
  0.03 &
   &
  1.06 &
  $\pm$ &
  0.01 &
   &
  0.01 \\
 &
  0.3 &
   &
  6 &
   &
  1.02 &
   &
  117 &
   &
  19 &
   &
  222 &
   &
  1.08 &
  $\pm$ &
  0.01 &
   &
  0.64 &
  $\pm$ &
  0.01 &
   &
  1.02 &
  $\pm$ &
  0.01 &
   &
  0.03 \\
 &
  0.3 &
   &
  6 &
   &
  10.2 &
   &
  254 &
   &
  65.3 &
   &
  130 &
   &
  1.10 &
  $\pm$ &
  0.02 &
   &
  0.82 &
  $\pm$ &
  0.01 &
   &
  1.01 &
  $\pm$ &
  0.00 &
   &
  0.03 \\
 &
  0.3 &
   &
  6 &
   &
  104 &
   &
  551 &
   &
  186 &
   &
  68.5 &
   &
  1.13 &
  $\pm$ &
  0.00 &
   &
  0.90 &
  $\pm$ &
  0.03 &
   &
  1.02 &
  $\pm$ &
  0.01 &
   &
  0.00 \\
 &
  0.3 &
   &
  10 &
   &
  0.05 &
   &
  43.1 &
   &
  4.93 &
   &
  509 &
   &
  0.86 &
  $\pm$ &
  0.09 &
   &
  0.18 &
  $\pm$ &
  0.01 &
   &
  1.10 &
  $\pm$ &
  0.01 &
   &
  0.01 \\
 &
  0.3 &
   &
  10 &
   &
  0.101 &
   &
  54.3 &
   &
  7.66 &
   &
  449 &
   &
  0.91 &
  $\pm$ &
  0.01 &
   &
  0.24 &
  $\pm$ &
  0.10 &
   &
  1.05 &
  $\pm$ &
  0.00 &
   &
  0.01 \\
 &
  0.3 &
   &
  10 &
   &
  1.01 &
   &
  117 &
   &
  30.2 &
   &
  281 &
   &
  1.06 &
  $\pm$ &
  0.02 &
   &
  0.63 &
  $\pm$ &
  0.01 &
   &
  0.99 &
  $\pm$ &
  0.00 &
   &
  0.03 \\
 &
  0.3 &
   &
  10 &
   &
  10.2 &
   &
  253 &
   &
  104 &
   &
  165 &
   &
  1.18 &
  $\pm$ &
  0.02 &
   &
  0.86 &
  $\pm$ &
  0.02 &
   &
  1.02 &
  $\pm$ &
  0.00 &
   &
  0.17 \\
 &
  0.3 &
   &
  10 &
   &
  103 &
   &
  549 &
   &
  301 &
   &
  87.6 &
   &
  1.13 &
  $\pm$ &
  0.02 &
   &
  0.87 &
  $\pm$ &
  0.03 &
   &
  1.02 &
  $\pm$ &
  0.00 &
   &
  0.00 \\
 &
  0.3 &
   &
  15 &
   &
  1.01 &
   &
  117 &
   &
  44.7 &
   &
  343 &
   &
  1.03 &
  $\pm$ &
  0.02 &
   &
  0.27 &
  $\pm$ &
  0.02 &
   &
  1.01 &
  $\pm$ &
  0.00 &
   &
  0.01 \\
 &
  0.3 &
   &
  15 &
   &
  10.2 &
   &
  253 &
   &
  155 &
   &
  200 &
   &
  1.06 &
  $\pm$ &
  0.01 &
   &
  0.55 &
  $\pm$ &
  0.02 &
   &
  1.03 &
  $\pm$ &
  0.00 &
   &
  0.02 \\
 &
  0.3 &
   &
  15 &
   &
  103 &
   &
  548 &
   &
  446 &
   &
  107 &
   &
  1.03 &
  $\pm$ &
  0.03 &
   &
  0.70 &
  $\pm$ &
  0.04 &
   &
  1.03 &
  $\pm$ &
  0.00 &
   &
  0.00 \\
 &
  1.5 &
   &
  2 &
   &
  0.05 &
   &
  43.1 &
   &
  6.06 &
   &
  564 &
   &
  1.38 &
  $\pm$ &
  0.01 &
   &
  0.65 &
  $\pm$ &
  0.02 &
   &
  1.04 &
  $\pm$ &
  0.00 &
   &
  0.00 \\
 &
  1.5 &
   &
  2 &
   &
  0.101 &
   &
  54.3 &
   &
  9.33 &
   &
  495 &
   &
  1.36 &
  $\pm$ &
  0.01 &
   &
  0.74 &
  $\pm$ &
  0.01 &
   &
  1.00 &
  $\pm$ &
  0.00 &
   &
  0.00 \\
 &
  1.5 &
   &
  2 &
   &
  1.01 &
   &
  117 &
   &
  36.7 &
   &
  310 &
   &
  1.36 &
  $\pm$ &
  0.01 &
   &
  0.94 &
  $\pm$ &
  0.00 &
   &
  1.00 &
  $\pm$ &
  0.00 &
   &
  0.01 \\
 &
  1.5 &
   &
  2 &
   &
  10.2 &
   &
  253 &
   &
  124 &
   &
  180 &
   &
  1.33 &
  $\pm$ &
  0.00 &
   &
  0.98 &
  $\pm$ &
  0.01 &
   &
  1.02 &
  $\pm$ &
  0.00 &
   &
  0.03 \\
 &
  1.5 &
   &
  2 &
   &
  103 &
   &
  547 &
   &
  347 &
   &
  94.5 &
   &
  1.36 &
  $\pm$ &
  0.00 &
   &
  1.02 &
  $\pm$ &
  0.03 &
   &
  1.01 &
  $\pm$ &
  0.00 &
   &
  0.00 \\
 &
  1.5 &
   &
  3 &
   &
  0.051 &
   &
  43.2 &
   &
  8.45 &
   &
  665 &
   &
  1.17 &
  $\pm$ &
  0.01 &
   &
  0.64 &
  $\pm$ &
  0.02 &
   &
  1.04 &
  $\pm$ &
  0.01 &
   &
  0.01 \\
 &
  1.5 &
   &
  3 &
   &
  0.101 &
   &
  54.4 &
   &
  12.9 &
   &
  582 &
   &
  1.15 &
  $\pm$ &
  0.01 &
   &
  0.72 &
  $\pm$ &
  0.15 &
   &
  0.96 &
  $\pm$ &
  0.01 &
   &
  0.02 \\
 &
  1.5 &
   &
  3 &
   &
  1.01 &
   &
  117 &
   &
  51 &
   &
  366 &
   &
  1.22 &
  $\pm$ &
  0.01 &
   &
  0.90 &
  $\pm$ &
  0.01 &
   &
  0.99 &
  $\pm$ &
  0.00 &
   &
  0.02 \\
 &
  1.5 &
   &
  3 &
   &
  10.2 &
   &
  253 &
   &
  174 &
   &
  213 &
   &
  1.26 &
  $\pm$ &
  0.01 &
   &
  0.96 &
  $\pm$ &
  0.03 &
   &
  1.03 &
  $\pm$ &
  0.00 &
   &
  0.02 \\
 &
  1.5 &
   &
  3 &
   &
  103 &
   &
  548 &
   &
  492 &
   &
  112 &
   &
  1.21 &
  $\pm$ &
  0.01 &
   &
  0.93 &
  $\pm$ &
  0.04 &
   &
  1.04 &
  $\pm$ &
  0.00 &
   &
  0.00 \\
 &
  1.5 &
   &
  6 &
   &
  0.051 &
   &
  43.3 &
   &
  15.7 &
   &
  902 &
   &
  1.35 &
  $\pm$ &
  0.05 &
   &
  0.30 &
  $\pm$ &
  0.03 &
   &
  1.05 &
  $\pm$ &
  0.01 &
   &
  0.01 \\
 &
  1.5 &
   &
  6 &
   &
  0.103 &
   &
  54.8 &
   &
  23.8 &
   &
  781 &
   &
  1.17 &
  $\pm$ &
  0.04 &
   &
  0.50 &
  $\pm$ &
  0.01 &
   &
  1.02 &
  $\pm$ &
  0.03 &
   &
  0.00 \\
 &
  1.5 &
   &
  6 &
   &
  1.01 &
   &
  117 &
   &
  94.3 &
   &
  498 &
   &
  1.15 &
  $\pm$ &
  0.01 &
   &
  0.80 &
  $\pm$ &
  0.03 &
   &
  1.03 &
  $\pm$ &
  0.00 &
   &
  0.01 \\
 &
  1.5 &
   &
  6 &
   &
  10.1 &
   &
  253 &
   &
  324 &
   &
  291 &
   &
  1.20 &
  $\pm$ &
  0.02 &
   &
  0.81 &
  $\pm$ &
  0.02 &
   &
  1.02 &
  $\pm$ &
  0.00 &
   &
  0.06 \\
 &
  1.5 &
   &
  6 &
   &
  102 &
   &
  546 &
   &
  930 &
   &
  155 &
   &
  1.17 &
  $\pm$ &
  0.03 &
   &
  0.93 &
  $\pm$ &
  0.02 &
   &
  1.04 &
  $\pm$ &
  0.00 &
   &
  0.00 \\
 &
  1.5 &
   &
  10 &
   &
  0.053 &
   &
  43.8 &
   &
  25.3 &
   &
  1130 &
   &
  1.22 &
  $\pm$ &
  0.05 &
   &
  0.14 &
  $\pm$ &
  0.05 &
   &
  1.03 &
  $\pm$ &
  0.00 &
   &
  0.01 \\
 &
  1.5 &
   &
  10 &
   &
  0.103 &
   &
  54.7 &
   &
  38.4 &
   &
  995 &
   &
  1.28 &
  $\pm$ &
  0.05 &
   &
  0.28 &
  $\pm$ &
  0.08 &
   &
  1.04 &
  $\pm$ &
  0.01 &
   &
  0.01 \\
 &
  1.5 &
   &
  10 &
   &
  1.02 &
   &
  118 &
   &
  153 &
   &
  630 &
   &
  1.10 &
  $\pm$ &
  0.05 &
   &
  0.71 &
  $\pm$ &
  0.06 &
   &
  1.03 &
  $\pm$ &
  0.00 &
   &
  0.01 \\
 &
  1.5 &
   &
  10 &
   &
  10.1 &
   &
  252 &
   &
  529 &
   &
  372 &
   &
  1.06 &
  $\pm$ &
  0.00 &
   &
  0.84 &
  $\pm$ &
  0.01 &
   &
  1.03 &
  $\pm$ &
  0.00 &
   &
  0.04 \\
 &
  1.5 &
   &
  15 &
   &
  1.01 &
   &
  546 &
   &
  222 &
   &
  762 &
   &
  1.25 &
  $\pm$ &
  0.00 &
   &
  0.67 &
  $\pm$ &
  0.04 &
   &
  1.04 &
  $\pm$ &
  0.00 &
   &
  0.01 \\
 &
  1.5 &
   &
  15 &
   &
  10.1 &
   &
  43.3 &
   &
  774 &
   &
  451 &
   &
  1.17 &
  $\pm$ &
  0.06 &
   &
  0.72 &
  $\pm$ &
  0.07 &
   &
  1.02 &
  $\pm$ &
  0.02 &
   &
  0.04 \\
 &
  4 &
   &
  2 &
   &
  0.051 &
   &
  54.4 &
   &
  15.7 &
   &
  904 &
   &
  \multicolumn{1}{l}{1.53} &
  $\pm$ &
  0.02 &
   &
  0.55 &
  $\pm$ &
  0.01 &
   &
  \multicolumn{1}{l}{1.03} &
  $\pm$ &
  0.00 &
   &
  0.01 \\
 &
  4 &
   &
  2 &
   &
  0.101 &
   &
  117 &
   &
  24.1 &
   &
  795 &
   &
  \multicolumn{1}{l}{1.39} &
  $\pm$ &
  0.01 &
   &
  0.67 &
  $\pm$ &
  0.03 &
   &
  \multicolumn{1}{l}{1.02} &
  $\pm$ &
  0.00 &
   &
  0.01 \\
 &
  4 &
   &
  2 &
   &
  1.01 &
   &
  253 &
   &
  96.4 &
   &
  503 &
   &
  \multicolumn{1}{l}{1.41} &
  $\pm$ &
  0.07 &
   &
  0.81 &
  $\pm$ &
  0.06 &
   &
  \multicolumn{1}{l}{1.02} &
  $\pm$ &
  0.00 &
   &
  0.01 \\
 &
  4 &
   &
  2 &
   &
  10.1 &
   &
  43.4 &
   &
  327 &
   &
  292 &
   &
  \multicolumn{1}{l}{1.31} &
  $\pm$ &
  0.04 &
   &
  0.89 &
  $\pm$ &
  0.01 &
   &
  \multicolumn{1}{l}{1.02} &
  $\pm$ &
  0.00 &
   &
  0.03 \\
 &
  4 &
   &
  3 &
   &
  0.051 &
   &
  54.8 &
   &
  22.2 &
   &
  1070 &
   &
  \multicolumn{1}{l}{1.35} &
  $\pm$ &
  0.04 &
   &
  0.52 &
  $\pm$ &
  0.04 &
   &
  \multicolumn{1}{l}{1.02} &
  $\pm$ &
  0.00 &
   &
  0.03 \\
 &
  4 &
   &
  3 &
   &
  0.103 &
   &
  118 &
   &
  33.8 &
   &
  930 &
   &
  \multicolumn{1}{l}{1.29} &
  $\pm$ &
  0.01 &
   &
  0.70 &
  $\pm$ &
  0.01 &
   &
  \multicolumn{1}{l}{1.02} &
  $\pm$ &
  0.00 &
   &
  0.04 \\
 &
  4 &
   &
  3 &
   &
  1.02 &
   &
  252 &
   &
  134 &
   &
  589 &
   &
  \multicolumn{1}{l}{1.26} &
  $\pm$ &
  0.01 &
   &
  0.84 &
  $\pm$ &
  0.02 &
   &
  \multicolumn{1}{l}{1.05} &
  $\pm$ &
  0.00 &
   &
  0.02 \\
 &
  4 &
   &
  3 &
   &
  10.1 &
   &
  117 &
   &
  461 &
   &
  348 &
   &
  \multicolumn{1}{l}{1.36} &
  $\pm$ &
  0.02 &
   &
  0.87 &
  $\pm$ &
  0.01 &
   &
  \multicolumn{1}{l}{1.02} &
  $\pm$ &
  0.00 &
   &
  0.02 \\
 &
  4 &
   &
  6 &
   &
  1.02 &
   &
  252 &
   &
  248 &
   &
  803 &
   &
  \multicolumn{1}{l}{1.24} &
  $\pm$ &
  0.01 &
   &
  0.76 &
  $\pm$ &
  0.01 &
   &
  \multicolumn{1}{l}{1.00} &
  $\pm$ &
  0.00 &
   &
  0.03 \\
 &
  4 &
   &
  6 &
   &
  10.1 &
   &
  43.3 &
   &
  858 &
   &
  475 &
   &
  \multicolumn{1}{l}{1.18} &
  $\pm$ &
  0.00 &
   &
  0.86 &
  $\pm$ &
  0.03 &
   &
  \multicolumn{1}{l}{1.00} &
  $\pm$ &
  0.00 &
   &
  0.05 \\
 &
  4 &
   &
  10 &
   &
  1.04 &
   &
  54.4 &
   &
  405 &
   &
  1020 &
   &
  \multicolumn{1}{l}{1.23} &
  $\pm$ &
  0.02 &
   &
  0.70 &
  $\pm$ &
  0.02 &
   &
  \multicolumn{1}{l}{1.01} &
  $\pm$ &
  0.00 &
   &
  0.02 \\
 &
  8 &
   &
  2 &
   &
  1.01 &
   &
  117 &
   &
  186 &
   &
  700 &
   &
  \multicolumn{1}{l}{1.56} &
  $\pm$ &
  0.01 &
   &
  0.46 &
  $\pm$ &
  0.02 &
   &
  \multicolumn{1}{l}{0.99} &
  $\pm$ &
  0.00 &
   &
  0.01 \\
 &
  8 &
   &
  2 &
   &
  10.1 &
   &
  252 &
   &
  640 &
   &
  410 &
   &
  \multicolumn{1}{l}{1.58} &
  $\pm$ &
  0.04 &
   &
  0.47 &
  $\pm$ &
  0.02 &
   &
  \multicolumn{1}{l}{0.95} &
  $\pm$ &
  0.00 &
   &
  0.02 \\ \hline
\end{tabular}%
}}
\end{table}

\newpage 

\section{Comparison of TCPMA System Results from Original and Swapped Arrangements\label{sec:swappedSI}}


The parity plots depicted in Figure S\ref{fig:Fig4-9} provide a visual comparison between the width factors and height factors obtained from the TCPMA system, at $Q=1.5$ LPM and $R_m=6$, for both the original and swapped arrangements. Each data point on the plot represents the average of three measurements at the given CPMA configurations. The parity line on the plot represents a metric where the factors obtained from the original and swapped CPMA arrangements would be identical. Any deviation from the parity line indicates a difference between the two arrangements. It is observed that the data points representing the width and height factors obtained from both the original and swapped CPMA configurations are clustered around the parity line, deviating by no more than approximately $10\%$ and $20\%$ (indicated by a shaded region in gray), respectively. This suggests that the two CPMAs evaluated have similar classification performance.

\begin{figure}[!htbp]
\includegraphics[scale=0.35]{FigS1.pdf} %max size={\textwidth}{\textheight}
\centering
      \caption{
        Comparison between the (a) width factors and (b) height factors derived from the original arrangement of TCPMA system and the swapped arrangement at a particular measurement configuration of $Q=1.5$ LPM and $R_m=6$.}\label{fig:Fig4-9}
     
  \centering
\end{figure}

Table S\ref{tab:tableG} provides a concise summary of the TCPMA experimental results obtained from the swapped CPMA arrangement, along with the corresponding operational values. 





 



\begin{table}
\centering
\caption{An overview of swapped tandem CPMA measurements as well as the fraction of positive multiply-charged particles exiting the first CPMA. The data presented in the table are expressed as mean values accompanied by 95\% confidence intervals.\label{tab:tableG}}
\textcolor{black}{\fontsize{30}{15}\selectfont
\resizebox{\columnwidth}{!}{%
\begin{tabular}{lclclclclcccllclclclclclcccc}
\multicolumn{26}{c}{Outline of TCPMA data (Swapped CPMA position)}\\                                                                                                                                                                                      & \multicolumn{1}{l}{} & \multicolumn{1}{l}{} \\ \hline \\
 & $Q$ & \multicolumn{1}{c}{} & $R_m$ & \multicolumn{1}{c}{} & $m^*$ &  & \multicolumn{1}{l}{$d_\mathrm{p}$} &  & $V$  & \multicolumn{1}{l}{} & $\omega$ & \multicolumn{1}{c}{} & \multicolumn{3}{c}{$\mu$} &  & \multicolumn{3}{c}{$\eta$} &  & \multicolumn{3}{c}{$m^*_{12}$} &  & \multicolumn{3}{c}{$F_{n_\mathrm{q}>1}$} \\
  & $_\mathrm{[LPM]}$ & \multicolumn{1}{c}{} &  & \multicolumn{1}{c}{} & $_\mathrm{[fg]}$ &  & \multicolumn{1}{l}{$_\mathrm{[nm]}$} &  & $_\mathrm{[V]}$  & \multicolumn{1}{l}{} & $_\mathrm{[rad/s]}$ & \multicolumn{1}{c}{} & \multicolumn{3}{c}{} &  & \multicolumn{3}{c}{} &  & \multicolumn{3}{c}{} &  & \multicolumn{3}{c}{} \\ \\
  \cline{2-2} \cline{4-4} \cline{6-6} \cline{8-8} \cline{10-10} \cline{12-12} \cline{14-16} \cline{18-20} \cline{22-24} \cline{26-28} 
  \\
& 1.5 & \multicolumn{1}{c}{} & 6     & \multicolumn{1}{c}{} & 0.052 &  & 43.3                               &  & 15.8 &                      & 907      & \multicolumn{1}{c}{} & 1.21   & $\pm$   & 0.010  &  & 0.369   & $\pm$   & 0.021  &  & 0.992    & $\pm$    & 0.017    &  & 0.00 & $\pm$                & 0.00                 \\\\
 & 1.5 &                      & 6     &                      & 0.102 &  & 54.8                               &  & 24   &                      & 785      &                      & 1.29   & $\pm$   & 0.027  &  & 0.443   & $\pm$   & 0.035  &  & 0.994    & $\pm$    & 0.007    &  & 0.00 & $\pm$                & 0.00                        \\\\
 & 1.5 &                      & 6     &                      & 1.01  &  & 117                                &  & 94.8 &                      & 500      &                      & 1.16   & $\pm$   & 0.002  &  & 0.685   & $\pm$   & 0.009  &  & 0.986    & $\pm$    & 0.002    &  & 0.01 & $\pm$                & 0.00                 \\\\
 & 1.5 &                      & 6     &                      & 10.1  &  & 253                                &  & 326  &                      & 293      &                      & 1.15   & $\pm$   & 0.012  &  & 0.747   & $\pm$   & 0.011  &  & 0.986    & $\pm$    & 0.001    &  & 0.04 & $\pm$                & 0.00                 \\\\
 & 1.5 &                      & 6     &                      & 102   &  & 546                                &  & 935  &                      & 156      &                      & 1.24   & $\pm$   & 0.010  &  & 0.935   & $\pm$   & 0.022  &  & 0.987    & $\pm$    & 0.003    &  & 0.00 & $\pm$                & 0.00                 
 \\\\ \hline
\end{tabular}%
}}
\end{table}


\section{Effective density of particle}
\label{sec:Effective_density}
The present section provides a theoretical investigation of the impact of variations in the effective density of particles on the width factor at various CPMA flow rates, mass set points, and resolutions. Figure S\ref{fig:Fig4A} displays the relative difference of the width factor obtained by convolving diffusion models for three different effective densities of particles. In particular, the relative differences in the width factor were determined for two particle densities of $\rho_{1}=500$ kg/m$^3$ and $\rho_{2}=2000$ kg/m$^3$ compared to the width factor calculated for effective density of $\rho_{0}=1198$ kg/m$^3$, as measured in the current study. The analysis was carried out utilizing three distinct resolutions (i.e., $R_m$= of 2, 6, and 15) for various flow rates ($Q$) ranging from 0.3 to 4 LPM and constrained to the operating region of Cambustion’s commercial CPMA.
\begin{figure}[!htbp]
\includegraphics[scale=0.32]{FigS2.pdf} %max size={\textwidth}{\textheight}
\centering
      \caption{
        This figure showcases the relative difference of the width factor, obtained by convolving diffusion models for three distinct effective densities of particles. Specifically, the absolute relative differences in the width factor were computed for $\rho_{1}=500$ kg/m$^3$ and $\rho_{2}=2000$ kg/m$^3$ compared to the width factor of the effective density of $\rho_{0}=1198$ kg/m$^3$, as measured in this study. The theoretical width factor was derived for various flow rates ($Q$) spanning 0.3 to 4 LPM and allowable operating region of CPMA, using three different resolutions ($R_m$) of 2, 6, and 15.}\label{fig:Fig4A}
     
  \centering
\end{figure}

The results suggest that changing the effective density of particles within the range of 500 to 2000 kg/m$^3$ has a significant effect on the width factor at low flows ($Q$=0.3 LPM), low resolutions ($R_m$=2), and low particle masses ($<$ 0.01 fg). However, practically speaking, the transmission efficiency of the CPMA is nearly zero in that range (see Figure 8a). Thus, the effect of particle effective density has a small effect on the width factor in the practical operating region of the CPMA.

\clearpage
% \newpage

 \bibliography{cas-refs}
\end{document}

\endinput





