**Research article**| 25 Nov 2022

# Survival probabilities of atmospheric particles: comparison based on theory, cluster population simulations, and observations in Beijing

Santeri Tuovinen Runlong Cai Veli-Matti Kerminen Jingkun Jiang Chao Yan Markku Kulmala and Jenni Kontkanen

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**Santeri Tuovinen et al.**Santeri Tuovinen Runlong Cai Veli-Matti Kerminen Jingkun Jiang Chao Yan Markku Kulmala and Jenni Kontkanen

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^{1}Institute for Atmospheric and Earth System Research/Physics, Faculty of Science, University of Helsinki, Helsinki, Finland^{2}State Key Joint Laboratory of Environment Simulation and Pollution Control, School of Environment, Tsinghua University, Beijing, China^{3}Joint International research Laboratory of Atmospheric and Earth System Research (JirLATEST), School of Atmospheric Sciences, Nanjing University, Nanjing, China^{4}CSC – IT Center for Science, Espoo, Finland

^{1}Institute for Atmospheric and Earth System Research/Physics, Faculty of Science, University of Helsinki, Helsinki, Finland^{2}State Key Joint Laboratory of Environment Simulation and Pollution Control, School of Environment, Tsinghua University, Beijing, China^{3}Joint International research Laboratory of Atmospheric and Earth System Research (JirLATEST), School of Atmospheric Sciences, Nanjing University, Nanjing, China^{4}CSC – IT Center for Science, Espoo, Finland

**Correspondence**: Markku Kulmala (markku.kulmala@helsinki.fi)

**Correspondence**: Markku Kulmala (markku.kulmala@helsinki.fi)

Received: 06 Jul 2022 – Discussion started: 05 Aug 2022 – Revised: 27 Oct 2022 – Accepted: 08 Nov 2022 – Published: 25 Nov 2022

Atmospheric new particle formation (NPF) events are regularly observed in urban Beijing, despite high concentrations of background particles which, based on theory, should inhibit NPF due to high values of coagulation sink (CoagS). The survival probability, which depends on both CoagS and particle growth rate (GR), is a key parameter in determining the occurrence of NPF events as it describes the fraction of newly formed particles that survive from a smaller diameter to a larger diameter. In this study, we investigate and compare survival probabilities from 1.5 to 3 nm (${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$), from 3 to 6 nm (${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$), and from 6 to 10 nm (${J}_{\mathrm{10}}/{J}_{\mathrm{6}}$) based on analytical formulae, cluster population simulations, and atmospheric observations from Beijing. We find that survival probabilities based on the cluster population simulations and one of the analytical formulae are in a good agreement. However, at low ratios between the background condensation sink (CS) and GR, and at high concentrations of sub-3 nm clusters, cluster–cluster collisions efficiently lower survival probabilities in the cluster population simulations. Due to the large concentrations of clusters and small particles required to considerably affect the survival probabilities, we consider it unlikely that cluster–cluster collisions significantly affect atmospheric survival probabilities. The values of ${J}_{\mathrm{10}}/{J}_{\mathrm{6}}$ observed in Beijing show high variability, most likely due to influences of primary particle emissions, but are on average in relatively good agreement with the values based on the simulations and the analytical formulae. The observed values of ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$ are mostly lower than those predicted based on the simulations and the analytical formulae, which could be explained by uncertainties in CS and GR. The observed values of ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ at high CS $/$ GR are much higher than predicted based on the simulations and the analytical formulae. We argue that uncertainties in GR or CS are unlikely to solely explain the observed values of ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ under high CS conditions. Thus, further work is needed to better understand the factors influencing survival probabilities of sub-3 nm atmospheric particles in polluted environments.

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Atmospheric new particle formation (NPF), consisting of the formation of stable clusters and their subsequent growth to larger sizes by the condensation of precursor vapors, has been frequently observed in many different environments (Kerminen et al., 2018). Aerosol particles affect both climate and human health (Pöschl, 2005; Rosenfeld et al., 2014; Shiraiwa et al., 2017; Bellouin et al., 2020) and NPF events significantly contribute to atmospheric concentrations of aerosol particles (Spracklen et al., 2010). Thus, NPF events can influence the effects of aerosol particles on climate and health. For example, they can increase the cloud condensation nuclei concentrations, thereby influencing climate and other properties of clouds (Spracklen et al., 2008; Yue et al., 2011). In addition, NPF events can contribute to haze episodes and lead to a degrading air quality (Guo et al., 2014; Kulmala et al., 2021, 2022).

The survival probability of molecular clusters and small aerosol particles
is one of the key parameters that determine whether an NPF event occurs or
not. It also determines the fraction of newly formed clusters, which are
eventually able to contribute to the preexisting particle population and
thereby potentially affect haze and aerosol–cloud interactions. In practice,
the survival probability describes the fraction of particles or clusters
formed at diameter *d*_{1} that grow to a larger diameter *d*_{2}. It can be
determined as the ratio of the formation rates of particles of diameters
*d*_{1} and *d*_{2} (Pierce and Adams, 2007; Kulmala et al., 2017). The
survival probability is governed by the growth rate (GR) and loss rate of
small particles and clusters (Kerminen and Kulmala, 2002; Kulmala et al.,
2017). GR depends on the concentrations of condensable precursor vapors
(Kulmala et al., 2005; Sihto et al., 2006; Wang et al., 2015; Stolzenburg et
al., 2020) and population dynamics such as cluster–cluster collisions
(Kontkanen et al., 2022). In addition, chemical reactions in the particle
may affect GR (Apsokardu and Johnston, 2018; Kulmala et al., 2022). The
losses of atmospheric new particles can be characterized by the coagulation
sink, denoted by “CoagS”, which describes the loss rate of small particles to
larger particles by coagulation (Dal Maso et al., 2002; Kulmala et al.,
2001). The value of CoagS depends on the diameter of the particle and is
higher for smaller particles (Kulmala et al., 2001; Dal Maso et al., 2002).
Condensation sink, denoted by “CS”, describes the loss rate of condensable
vapor, often sulfuric acid, on particles, and it is frequently used as a proxy
for the CoagS (Dal Maso et al., 2002; Kerminen and Kulmala, 2002). If the
ratio of CoagS, or CS, to GR is low, the survival probability is high and
high fractions of small particles are able to survive to larger sizes
(Kulmala et al., 2017). This can result in an NPF event being observed if
the initial concentrations of clusters are sufficiently high.

In this study, we focus on the survival of new particles in the polluted
atmosphere in Beijing, China, where NPF events have been observed to occur
frequently despite the high CoagS (Chu et al., 2019; Deng et al., 2020; Cai
et al., 2021b). The median CS during NPF days in Beijing was found to be
∼0.02 s^{−1} (Deng et al., 2020; Wu et al., 2008; Zhou et al., 2020). Previous studies have shown that the
survival probabilities in Beijing and other megacities are significantly
higher than theoretically predicted under high CoagS and CS conditions
(Kulmala et al., 2017; Yao et al., 2018). For example, in Beijing, NPF events
have been observed to occur even when the ratio of CS to GR is so high
(e.g., >50 nm^{−1}) that theoretically no formation of 3 nm
particles should be possible (Kulmala et al., 2017). This indicates a gap in
the current understanding of NPF and this is why the survival probabilities
of small particles in polluted environments are of great interest.

Here, we will investigate survival probabilities in the diameter ranges 1.5–3, 3–6, and 6–10 nm based on observations from Beijing, cluster population simulations using Atmospheric Cluster Dynamics Code (ACDC) (McGrath et al., 2012; Olenius and Riipinen, 2017), and analytical formulae. Three different size ranges of the survival probability will be investigated, as motivated by the size dependency of the survival probability (Kerminen and Kulmala, 2002) in order to find out whether there is an agreement between the observations, simulations, and analytical formulae, and how this depends on the size range. The observations are based on particle number size distribution measurements from 1 year (2018), from which CS, GR, and the formation rates of 1.5, 3, 6, and 10 nm particles will be derived. The formation rates will then be used to determine survival probabilities. Cluster population simulations make it possible to consider the effects of particle population dynamics on survival probabilities (Kontkanen et al., 2022). Two different sets of simulations will be conducted: first, assuming that there are no collisions between newly formed particles (i.e., particles grow only by collisions of vapor molecules) and then allowing collisions between new particles to occur. Theoretical predictions for survival probabilities are determined according to analytical formulae by Kerminen and Kulmala (2002), Lehtinen et al. (2007), and Korhonen et al. (2014). These equations relate the formation rate at a larger diameter to the formation rate at a smaller diameter, and since survival probability can be expressed as a ratio of formation rates, they can be used to determine survival probabilities from a smaller diameter to a larger diameter. The difference between the equations by Lehtinen et al. (2007) and Korhonen et al. (2014) is that the former assumes a constant GR while the latter assumes a linear or a power-law dependency of the GR on the particle diameter. The Kerminen and Kulmala (2002) equation also assumes a constant GR; however, it differs from the equation by Lehtinen et al. (2007) by relying on CS instead of CoagS and by handling the size dependency of the sink term differently.

The main objectives of this study are (1) to investigate if, and how, the survival probabilities of sub-10 nm particles differ between observations, cluster population simulations, and analytical formulae; (2) to evaluate the effect of uncertainties in the observed parameters on our results; and to (3) discuss other possible explanations, such as ineffective CoagS or enhanced GR. In practice, we will first compare theoretical survival probabilities, the survival probabilities from cluster population simulations, both with and without cluster–cluster collisions, and the observed survival probabilities. Then we will investigate how large of an uncertainty in GR or CS is needed to explain the observed survival probabilities and consider some reasons for the inaccuracy of CoagS and GR, including the measurement uncertainty and some assumptions made in determining CoagS or GR. Finally, other explanations for our results will be briefly discussed. Based on our results, we can get a better understanding of NPF in polluted megacities and gain more insight into the reasons behind disagreements between the predicted and observed survival probabilities.

## 2.1 Theoretical survival probability

The survival probability of atmospheric particles or clusters describes the
probability, or fraction, of particles of a smaller diameter *d*_{1} growing
to a larger diameter *d*_{2} (Pierce and Adams, 2007). In other words, it is
the probability that a growing particle, initially of diameter *d*_{1}, is
not lost due to coagulation scavenging and other loss mechanisms, such as
dry deposition, before it reaches the diameter *d*_{2}. Coagulation
scavenging, described by the CoagS, is usually the most
important sink for sub-10 nm particles, and in this study, it is the only
loss mechanism of particles we consider.

The best method to determine the survival probability from observations
depends on particle size distribution and its temporal evolution (Cai et al.,
2022). In this study, we have determined the survival probability as a ratio
of formation rates *J*_{1} and *J*_{2} (Kerminen and Kulmala, 2002; Kulmala
et al., 2017). This method is able to produce accurate survival
probabilities for a steady-state or a quasi-steady-state size distribution,
and has been shown to give relatively accurate survival probabilities in
Beijing during NPF events (Cai et al., 2022).

In this study we consider three different survival probabilities: from
1.5 to 3 nm (${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$), from 3 to 6 nm (${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$), and from 6 to 10 nm (${J}_{\mathrm{10}}/{J}_{\mathrm{6}}$). Based on Kerminen and Kulmala (2002), the
survival probability against coagulation scavenging from *d*_{1} to *d*_{2}
is

where GR is the growth rate between *d*_{1} and *d*_{2} and CS${}^{\prime}=\text{CS}/\left(\mathrm{4}\mathit{\pi}D\right)$, CS is the condensation sink of sulfuric acid, and *D* is the
diffusion coefficient of sulfuric acid. CS is used as a proxy for the CoagS. Parameter *η* is a semi-empirically derived
quantity taking into account the influence of the background particle size
distribution on the size dependency of the CoagS. It is approximately equal
to 0.23 nm^{2} m^{2} h^{−1}.

The survival probability can also be determined based on the formulations by Lehtinen et al. (2007) and Korhonen et al. (2014), which directly use the CoagS instead of the CS to describe the particle scavenging losses. Based on these two studies, ${J}_{\mathrm{2}}/{J}_{\mathrm{1}}$ can be written as

where CoagS_{1} is the coagulation sink at the smaller diameter
*d*_{1}. If ${J}_{\mathrm{2}}/{J}_{\mathrm{1}}$ is determined based on Lehtinen et al. (2007), the
growth rate (GR) is assumed to be constant, GR = GR_{1−2}, and GR_{1−2}
is the growth rate from the smaller diameter *d*_{1} to the larger diameter
*d*_{2}. In this case

Here, the parameter *m* depends on the background particle distribution, and it
is defined as

CoagS_{2} is the coagulation sink of the larger particle with diameter
*d*_{2}. In this work, if not otherwise stated, we assume that $m=-\mathrm{1.6}$ when
making predictions for the survival probabilities based on the analytical
formulae. The effects of this assumption will be considered in Sect. 3.2,
where the sensitivity of the survival probability on *m* is briefly
investigated.

If the survival probability is determined based on Korhonen et al. (2014),
GR is either assumed to have a linear or power-law size dependency on the
particle diameter. In this study, we only considered the case with power-law
size dependency, in which case GR in Eq. (2) is the growth rate at the smaller
size *d*_{1}, GR = GR_{1}, and

The parameter *n*, related to the size dependency of GR, is analogous to the
parameter *m*, and it is defined as

where GR_{2} is the growth rate at diameter *d*_{2}.

From here on, we will refer to the predicted survival probabilities based on Kerminen and Kulmala (2002), Lehtinen et al. (2007), and Korhonen et al. (2014) with KK-2002, L-2007, and K-2014, respectively.

## 2.2 Formation rate of atmospheric clusters and particles

The particle formation rate (*J*_{i}) is one of the parameters used to
characterize NPF. It describes the flux of growing particles past some
diameter *d*_{i}. Using the formulation derived on the basis of the aerosol general
dynamic equation, *J*_{i} can be determined from the particle number size
distribution using (Kulmala et al., 2012)

Here,${N}_{[{d}_{i},{d}_{u})}$is the particle number concentration between
diameters *d*_{i} and *d*_{u}, excluding the upper limit *d*_{u}.
${\text{GR}}_{[{d}_{i},{d}_{u})}$ is the growth rate and CoagS is the coagulation sink
of the particles in the size range. To account for the coagulation effects
and their influence on *J*_{i} more accurately, *J*_{i} can be calculated based
on an improved formulation of Eq. (7) proposed by Cai and Jiang (2017):

Here, *d*_{j} is the lower limit of *j*th measured size bin,
*K*_{(j,g)} is the coagulation coefficient for collisions between
particles with diameters *d*_{j} and *d*_{g}, *d*_{min} is the lowest
measured particle diameter, *n*_{u} is the particle size distribution
function, and GR_{u} is the growth rate at *d*_{u}.

## 2.3 Growth rate and condensation and coagulation sinks

The particle GR describes the rate of change of particle
diameter with time. In this study, the value of GR was determined using the
appearance time method (Lehtipalo et al., 2014; Olenius et al., 2014). This
method is based on finding a corresponding time of appearance
(*t*_{app,i}) for each particle size bin *i*, usually defined as the time
that the number concentration in that size bin reaches 50 % of its maximum
value during an NPF event. The GR is then estimated using the slope of the
diameters versus the corresponding *t*_{app}

GR derived from the appearance times can be affected by processes other than the particle growth such as coagulation scavenging, which can lead to an overestimation of GR (Stolzenburg et al., 2005; Cai et al., 2021a). Thus, in this work GR was corrected for the influence of background CoagS according to the procedure presented by Cai et al. (2021a).

CS, which describes the loss rate of condensing vapor to background particles, can be determined from the particle number size distribution (Dal Maso et al., 2002)

where *β*_{j} is the transition regime correction factor (Fuchs and
Sutugin, 1971). CoagS_{i}, which describes the loss rate of small
particles of diameter *d*_{i} to larger background particles, is obtained
from the following equation:

where *K*_{i,j} is the coagulation coefficient between collisions of the
particles *i* and *j*. Due to the similar dependency on the particle number size
distribution, CoagS can be determined from the CS using the parameter *m*,
which was introduced in Sect. 2.1. Thus,

where *d*_{mon} is the diameter of the condensing monomer. Because of this
relation between the CS and the CoagS, and the size dependency of the CoagS,
we have chosen to use CS $/$ GR (nm^{−1}), where the CS is for sulfuric acid,
to represent the ratio between coagulation scavenging and particle growth.
The survival probabilities in this study will thus be presented with respect
to CS $/$ GR.

## 2.4 Cluster population simulations

We investigated the agreement between survival probabilities from analytical formulae, atmospheric observations, and cluster population simulations using Atmospheric Cluster Dynamics Code (ACDC). These simulations were used as an intermediate step between the theory and the atmospheric observations. In addition, they provide valuable information on the agreement between the survival probabilities from analytical predictions and cluster population simulations, which to our knowledge have not been published before.

The ACDC program models the first steps of the atmospheric cluster and particle formation by solving the aerosol general dynamic equation (McGrath et al., 2012; Olenius and Riipinen, 2017):

where *K*_{i,j} is the coagulation coefficient between collisions of clusters
or *i* and *j*, *E*_{i→j} is the evaporation coefficient from the cluster
*i* to two smaller clusters, the other of which is the cluster *j*. *Q*_{i}
describes external sources of clusters *i* while *S*_{i} describes their external
losses, such as wall losses, other than CoagS_{i}, which describes the
coagulation losses of clusters to a background particle population. In our
simulations *Q*_{i} and *S*_{i} were set to 0. The coagulation coefficients
were determined assuming hard sphere collisions with an accommodation
(sticking) factor of unity. Note that in this study particles and clusters
are referred to as clusters, regardless of their size, when concerning the
cluster population simulations.

We assumed monomer of the model substance to have properties corresponding
to a cluster consisting of one sulfuric acid and one dimethylamine molecule,
similar to the approach by Kontkanen et al. (2018). This corresponds to a
situation where every sulfuric acid molecule is bound to a dimethylamine
molecule. Sulfuric acid and dimethylamine cluster formation has been
observed to be the main pathway of the initial formation of atmospheric
clusters in Beijing (Cai et al., 2021b). Dimethylamine effectively
stabilizes sulfuric acid clusters, if its concentration is sufficient with
respect to the atmospheric sulfuric acid concentration (Jen et al., 2014;
Kürten et al., 2014). We assumed that the formation of clusters occurred
at a kinetic limit, which means that evaporation from clusters is negligible
and cluster formation and growth are governed by kinetic collisions. Thus,
in Eq. (13) *E*=0.

Two different simulation sets were considered. In the first simulation set,
later referred to as Case 1, collisions only occurred between the
monomers and the clusters, which means that in Eq. (13) ${K}_{i,j}\ne \mathrm{0}$,
only when the cluster *j* is the monomer (*j*=1). In the second simulation
set, referred to as Case 2, collisions between the clusters were also allowed
to occur. Thus, while in Case 1 the cluster growth was only due to
condensation, in Case 2 smaller clusters also contributed to the growth and,
at the same time, larger clusters contributed to the losses of the smaller
clusters.

The monomer concentration (*C*_{mon}) was kept constant during all the
simulations at ${C}_{\mathrm{mon}}=\mathrm{1}\times {\mathrm{10}}^{\mathrm{7}}$ cm^{−3}, the value of
which was chosen based on previous studies of sulfuric acid concentrations
in Beijing (Yue et al., 2010; Lu et al., 2019; Li et al., 2020). We note
that while the lifetime of sulfuric acid in Beijing is short due to high CS,
sulfuric acid concentration can be assumed to be relatively constant during
the time it takes, for example, for a 1.5 nm particle to grow to 3 nm. The
largest modeled clusters in the simulations consisted of 4000 monomers and
were over 10 nm in diameter. The background CS of the monomer, which was
given as an input to the program, was varied between $\mathrm{5}\times {\mathrm{10}}^{-\mathrm{4}}$ and 0.02 s^{−1} within the simulation sets. The model calculated
the corresponding background CoagS of clusters based on Eq. (12). The
properties of the monomer and other constant properties are presented in
Table 1. We note that while the typical CS in Beijing is much higher than
the lowest CS values used in this study, we have selected them so that the
resulting CS $/$ GR values are in a comparable range with the observed values.

ACDC was run until steady state, i.e., until the concentrations of clusters
with diameters up to 10 nm did not considerably change with time anymore
($\mathrm{d}{N}_{i}/\mathrm{d}t\approx \mathrm{0}$). The simulated steady-state size
distributions for both Case 1 and Case 2 are presented in Figs. S2 and S3 in the Supplement. The cluster formation rates at the diameters 1.5, 3, 6,
and 10 nm, which we needed to determine the survival probabilities, were
returned by the program as the cluster flux past these sizes. The cluster
fluxes were determined by the program based on the cluster and monomer
concentrations and the collision rates between the different clusters or
between the clusters and the monomers. In addition, formation rates were
also determined based on Eq. (7). Steady-state formation rates from Eq. (7) were
determined using the concentrations in different size bins, which the
program returned as an output file. CoagS was approximated by CoagS of a
cluster with a geometric mean diameter of the upper and lower limits of the
considered diameter range. Since the particle size distributions are at the
steady state, the change of the cluster concentration with time is 0 in
Eq. (7). The upper *d*_{u} limit for *J*_{1.5}, *J*_{3}, and *J*_{6} is 10 nm, and
for *J*_{10} it is 10.7 nm, corresponding to the largest clusters with
non-zero number concentrations in the system.

The GR was determined based on the appearance time method (see Sect. 2.3). Both polynomial regression and linear regression with three size ranges (1.5–3, 3–6, 6–10 nm) were used to determine GR from the appearance times. Unless otherwise stated, the values of GR presented in this study are based on polynomial regression due to the strong size dependency of GR, and if a constant GR is used, such as for the ratio CS $/$ GR, it is based on arithmetic mean GR.

## 2.5 Measured survival probability in Beijing

In this work we used measured particle number size distributions and
measurement-based values of CS, GR, and formation rates from Beijing to
determine the survival probabilities ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$, ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$, and
${J}_{\mathrm{10}}/{J}_{\mathrm{6}}$ and the corresponding ratios between CS and GR. All the data
were based on measurements at the station of Beijing University of Chemical
Technology (39^{∘}56^{′}31^{′′} N, 116^{∘}17^{′}50^{′′} E, Beijing) during
2018. The particle number size distribution between 1 nm and 1 µm was
measured with a diethylene glycol scanning mobility particle sizer and a
custom-made particle size distribution system (Jiang et al., 2011; Liu et
al., 2016; Cai et al., 2017a). The formation rates for 1.5 nm (*J*_{1.5}), 3 nm (*J*_{3}), 6 nm (*J*_{6}), and 10 nm (*J*_{10}) particles were determined
using Eq. (8). The upper limit for the determination of formation rate,
*d*_{u}, was 3 nm for *J*_{1.5} and 25 nm for the other *J*. More details on the
measurement site and measurements can be found in Zhou et al. (2020).

Days were classified as NPF event days if a new mode below 25 nm appeared
and its growth to larger sizes was observed within the following hours (Dal
Maso et al., 2005). Only the NPF event days with a clear appearance of sub-3 nm particles and growth up to over 10 nm sizes were included in the
analysis. We determined the formation rates for survival probability
calculation with a time delay in order to account for the non-steady-state
aerosol size distributions during NPF. The time delay between the formation
of particles of diameter *d*_{1} and particles of diameter *d*_{2} was
determined based on the GR between these two sizes:

This time delay was then used to choose the formation rate at *d*_{2}
(*J*_{2}) corresponding to formation rate at the diameter *d*_{1}(*J*_{1}). Thus, survival probability from *d*_{1} to *d*_{2} is

The value of CS for the ratio CS $/$ GR corresponding to the survival
probability was calculated as the median value between *t*_{1} and *t*_{2}. All
the times were chosen so that the earliest and the last *J*_{1.5} value from
the day corresponded to the approximate start and end of formation of 1.5 nm
particles, which were based on a visual analysis of the event day particle
number size distributions. We used only one daily GR value for a certain
size range and assumed that throughout the day GR from a smaller size
*d*_{1} to a larger size *d*_{2} remains the same. In addition to inaccuracies
in CS $/$ GR, this assumption also affects the values of the survival
probabilities themselves as we have used the GR to select the formation
rates. The atmospheric particle GR can be expected to vary, for example, due
to changes in the concentrations of different condensing vapors.

The values of GR were determined on the basis of the appearance time method (see
Sect. 2.3) and they were determined based on linear regression for limits 1.5–3 nm (GR_{1.5−3}), 3–7 nm (GR_{3−7}), and 7–25 nm
(GR_{7−25}). For calculating the survival probabilities
${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ and ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$, GR_{1.5−3} and GR_{3−7} were used,
respectively, whereas for calculating ${J}_{\mathrm{10}}/{J}_{\mathrm{6}}$, the weighted mean
value of GR_{3−7} and GR_{7−25} was used. These same values of GR
were also used to determine CS $/$ GR corresponding to the survival
probabilities.

## 3.1 Formation rates and growth rates in cluster population simulations

Figure 1 shows GRs based on multi-degree
polynomial regression from the cluster population simulations for both Case 1 and Case 2 for different input background CS. In Case 1 collisions between
clusters did not occur while in Case 2 they did. In Case 1, the GR has very
similar values at all CS values since growth only occurs through
condensation and *C*_{mon} is constant. GR depends on *d*_{i} so that smaller
*d*_{i} has a larger GR. This size dependency is stronger at smaller
diameters, while for larger *d*_{i} GR is almost constant. This observed
behavior of GR in the simulations for Case 1 results from the coalescence
with the monomer increasing the size of a smaller cluster relatively more
than that of a larger cluster (Nieminen et al., 2010).

Figure 1 also shows the GR for Case 2. When
including cluster–cluster collisions, GR as a function of *d*_{i} shows very
different behavior depending on the background CS. When CS is small, GR is
much higher for all the cluster sizes compared to Case 1. With increasing
CS, GR becomes smaller, with the change being larger for larger *d*_{i}. If
CS = 0.02 s^{−1}, GR is significantly larger for Case 2 compared to Case 1
at small *d*_{i}, the difference being over 1 nm h^{−1} at *d*_{i}=1.5 nm. With
the same background CS, the difference in GR for the two cases is
approximately 0.2 nm h^{−1} at *d*_{i}>3 nm. In Case 2 the GR depends
strongly on background CS since at smaller background CS, the concentrations
of clusters are higher and the cluster–cluster collisions contribute more to
the growth. On the contrary, when the background CS is larger, the cluster
concentrations remain small and the effect of cluster–cluster collisions on
the cluster growth remains minor. The cluster GR can be significantly
enhanced by cluster–cluster collisions if the cluster concentrations are
sufficiently high in comparison with CS. Lehtipalo et al. (2016) have
previously shown that if the concentrations of stabilizing vapors such as
dimethylamine are high, resulting in low or negligible evaporation rates,
cluster–cluster collisions can have a major contribution to the growth of
clusters and particles, especially if CS is low. The same has also been
observed in studies utilizing cluster population simulations (Kontkanen et
al., 2022).

It should be noted that while we do not explicitly account for the effect of cluster–cluster collisions on the coagulation losses of the clusters in this study, the loss rates of the clusters can also be considerably affected by the cluster–cluster collisions if the background CS is small and cluster concentrations are high.

ACDC determines the output formation rates directly based on the cluster growth fluxes over the threshold diameters, whereas the formation rates from the atmospheric observations are determined based on the measured particle number size distribution (see Sect. 2.2). To investigate whether this difference in method to determine the formation rate could result in disagreements between the cluster population simulations and the observations, we compared the cluster population model steady-state formation rates from the fluxes with the formation rates calculated on the basis of Eq. (7) for both Case 1 and Case 2.

For Case 1, Fig. 2 shows that for all
the sizes, the two formation rates for Case 1 are approximately within a
factor of 2. *J*_{3}, *J*_{6}, and *J*_{10} are very close to the same value
despite the method used for its determination. However, the majority of the
values of *J*_{1.5} are smaller if the formation rates are determined
based on Eq. (7) compared to if they are based on the fluxes, which we assume
to be caused by the mean GR between 1.5 and 10 nm underestimating the
growth slightly. Thus, whether we determine formation rate based on fluxes
or Eq. (7) does not cause a significant difference in the values of
${J}_{\mathrm{10}}/{J}_{\mathrm{6}}$ or ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$ for Case 1. Determining ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ based on
the formation rates from Eq. (7) results in a larger value than if the
formation rates are directly based on the cluster growth fluxes. However,
the differences are relatively minor. For Case 2, Fig. 2 shows that when
the formation rates are high, and CS is low, the formation rates based on
Eq. (7) are lower than those based on fluxes. This is because Eq. (7) uses only
one value of CoagS to approximate the coagulation losses between the upper
and lower limits of the diameter range. The CoagS that is used underestimates these
losses because in Case 2 other clusters contribute to the losses of clusters
in addition to the background CS. As the observed formation rates are based
on Eq. (8), which includes a more detailed description of coagulation
scavenging compared with Eq. (7), we assume that the observed survival
probabilities are comparable to both Case 1 and Case 2 survival
probabilities.

## 3.2 Sensitivity of survival probability to CS $/$ GR and *m*-parameter

We investigated the sensitivity of L-2007 survival probability to the ratio
CS $/$ GR and the parameter *m*. Figure 3
illustrates the sensitivity of the survival probabilities to CS $/$ GR and shows
how uncertainties in CS, or GR, can lead to disagreements between the
observed survival probability and theoretical survival probability. For
example, if CS $/$ GR is 40, ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ can be underestimated by almost
3 orders of magnitude if the true CS $/$ GR is 50 % of the assumed value.
Similarly, if the true CS $/$ GR is 50 % larger than the assumed CS $/$ GR of 40,
we can overestimate ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ by 2 orders of magnitude. The survival
probability is less sensitive to CS $/$ GR when CS $/$ GR is low, and thus the error
in CS $/$ GR results in a larger error in the survival probability at high
CS $/$ GR. If either the GR or CS determined based on measurements is
inaccurate, the predicted theoretical survival probability can be
significantly different from the one we observe. This is especially true for
highly polluted environments where CS $/$ GR is often quite high.

Additionally, we investigated the sensitivity of L-2007 survival probability
to the parameter *m* (Fig. 4).
${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$ and ${J}_{\mathrm{10}}/{J}_{\mathrm{6}}$ are more sensitive to *m* compared with
${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$. The parameter *m* varies to some extent as the number size
distribution of larger particles changes, which can also affect the survival
probability of particles or clusters. For example, when CS $/$ GR is around 20,
${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$ is approximately 0.02, 0.05, and 0.08 when $m=-\mathrm{1.4}$, −1.5, and
−1.6, respectively. However, considering the uncertainties associated with
the measured formation rates, CS and GR, we may assume that the effect of
our assumption that $m=-\mathrm{1.6}$ on our results is relatively minor. We discuss this further in Sect. 3.4.

## 3.3 Survival probabilities in cluster population model simulations

Figure 5 shows the survival probabilities ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$, ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$, and ${J}_{\mathrm{10}}/{J}_{\mathrm{6}}$ based on the cluster population model simulations and analytical formulae (see Sect. 2.1) as a function of CS $/$ GR. In addition, ${J}_{\mathrm{10}}/{J}_{\mathrm{1.5}}$ based on the cluster population model simulations is shown. Note that CS in CS $/$ GR is the background CS, which for the ACDC Case 1 corresponds to the total loss rate. The ratio CS $/$ GR corresponding to the values of survival probabilities from the cluster population model simulations was determined by the appearance time method based on multi-degree polynomial regression (see Sects. 2.3 and 2.4). Additionally, results with GR determined on the basis of linear regression are shown in Fig. S1.

We see that for Case 1 with no cluster–cluster collisions, the ratio CS $/$ GR corresponding to a value of the survival probability ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ is higher if GR is based on the linear regression of the appearance times (Fig. S1) than on multi-degree polynomial regression (Fig. 5), which is especially apparent at higher CS. As GR is highly size dependent at smaller diameters, determining it based on the linear fit leads to a higher uncertainty in GR between 1.5 and 3 nm. This underestimation of GR increases ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ at certain CS $/$ GR. While GR based on the linear regression of the appearance times is at most approximately 30 % lower than GR based on the multi-degree polynomial regression of the appearance times, the effect on the interpretation of our results is significant: at similar CS $/$ GR the value of ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ can seem to be much higher. Similar observations can be made about ACDC Case 2 with cluster–cluster collisions at high CS. As discussed in Sect. 3.2, survival probabilities are highly sensitive to uncertainties in CS and GR, and thus GR should always be determined as accurately as possible. For this reason, further discussion of the cluster population model survival probabilities is focused on the results with GR based on the multi-degree polynomial regression of the appearance times.

For Case 1 with no cluster–cluster collisions, we can see that ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ from the cluster population simulations is higher than the KK-2002 ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}},$; however, the difference is relatively small. ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$ and ${J}_{\mathrm{10}}/{J}_{\mathrm{6}}$ from the simulations are considerably lower than the KK-2002 ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$ and ${J}_{\mathrm{10}}/{J}_{\mathrm{6}}$. The L-2007 survival probabilities are closer to the cluster population simulation survival probabilities and the only notable, but still relatively small, differences are observed at high CS $/$ GR for ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ and ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$. The ACDC Case 1 ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ is higher than the L-2007 ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ by a bit more than a factor of 2 at its largest value.

The differences in the KK-2002 and the L-2007 survival probabilities are due to the formula by Kerminen and Kulmala (2002) having a less accurate size dependency of CoagS. This is because in the derivation of the formula by Kerminen and Kulmala (2002), a power-law dependency of CoagS with an exponent of −2 was assumed. The assumption is then corrected by a semi-empirically derived correction parameter, while the Lehtinen et al. (2007) formulation directly accounts for the size dependency of CoagS.

Both the KK-2002 and the L-2007 survival probabilities are determined using a constant GR, which in this case is the arithmetic mean GR in the relevant size range. Since GR for Case 1 in the cluster population simulations is highly size dependent (see Fig. 1), the small differences between the L-2007 and the ACDC Case 1 survival probabilities are probably explained by this assumption of a constant GR. This is supported by the K-2014 survival probabilities, which were determined assuming a power-law size dependency of GR, being approximately the survival probabilities from the cluster population simulations for the three size ranges. We note that as ACDC Case 1 survival probabilities show only minor differences compared to L-2007 survival probabilities; the mean GR in a size range appears to represent the growth term in CS $/$ GR well. Thus, we assume that the values of CS $/$ GR from ACDC model simulations are comparable to the observed values of CS $/$ GR.

Figure 5 shows ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$, ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$, and ${J}_{\mathrm{10}}/{J}_{\mathrm{6}}$ for ACDC Case 2. In Case 2, collisions between clusters occurred in addition to the collisions between the monomers and the clusters. We see that for larger CS $/$ GR, the behavior of the survival probabilities is similar to Case 1. However, for CS $/$ GR roughly below 20, ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$, ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$, and ${J}_{\mathrm{10}}/{J}_{\mathrm{6}}$ are considerably smaller than the survival probabilities based on the analytical formulae and from Case 1. The Case 2 ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ is smaller than the Case 1 ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ by more than 1 order of magnitude when CS $/$ GR <7. In addition, Fig. 5 shows ${J}_{\mathrm{10}}/{J}_{\mathrm{1.5}}$ for both Case 1 and Case 2. It is clear that the survival rates from 1.5 up to 10 nm are considerably decreased by cluster–cluster collisions when background CS is low. This is because when CS is low and the concentrations of clusters are high, cluster–cluster collisions reduce the cluster number concentrations more efficiently than they increase the survivability of clusters from coagulation scavenging due to enhanced growth. A similar decrease in survival probabilities due to high numbers of collisions between clusters has been previously shown in model simulations and CLOUD chamber experiments by Xiao et al. (2021).

Based on our results we cannot directly say whether, and to what extent, particle and cluster survival probabilities in atmospheric environments are affected by cluster–cluster collisions. However, we have shown that if the concentrations of sub-10 nm particles and clusters are high, the survival probabilities can be considerably influenced by the increased loss rates. In addition, the dependency of survival probability on the background CS can be weakened due to high rates of collisions between sub-10 nm particles and clusters.

## 3.4 Survival probabilities in Beijing

### 3.4.1 Survival probabilities during a median new particle formation event

In the following, we will consider a median NPF event day in Beijing (Fig. 6), by which we mean that the median
diurnal variation of values was determined by calculating median values for
each 10 min time interval based on the data from all the investigated dates.
The median formation rates and survival probabilities for the three
investigated size ranges are shown in Fig. 6 alongside the L-2007 survival
probabilities (see Sect. 2.1). The values of the median survival
probabilities and CS $/$ GR ratio were determined based on Eqs. (13) and (14).
During the median event, the GR between 1.5 and 3 nm is 2.5 nm h^{−1}, between 3 and
6 nm it is 4.9 nm h^{−1}, and between 6 and 10 nm it is 3.4 nm h^{−1}. Between 07:00
and 14:00 (local time) the CS $/$ GR ratio corresponding to the survival
probability ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ varies approximately between 8 and 11, while the
CS $/$ GR ratios corresponding to ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$ and ${J}_{\mathrm{10}}/{J}_{\mathrm{6}}$ vary approximately
between 4 and 6, and 6 and 8, respectively. The variability in CS $/$ GR during the median day
is quite minor; however, it should be noted that the variability during
separate days can be considerably larger.

Figure 6 illustrates how the formation rates and the survival probabilities
vary during an NPF event. At first the formation rates increase during the
event, reaching a peak after which they start to decrease. *J*_{1.5} reaches
the peak value first, followed by *J*_{3}, *J*_{6}, and *J*_{10} demonstrating
the time delay between the formation of different diameters. The value of CS
during the median NPF event varies little and because of the following low
variability of CS $/$ GR, the predicted survival probabilities are
relatively constant during the event. The median ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ is mostly
higher than the L-2007 ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ except in the afternoon when clusters
do not appear to grow to larger sizes effectively. However, the difference
is relatively small, less than by a factor of 3 at its highest. The
median ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$ is always lower than the L-2007 ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$ and most
of the time the difference is larger than for ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}.$ At its peak
value, when the difference is the smallest, the median ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$ is
∼70 % of the predicted ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$. When the median event is first observable, the median ${J}_{\mathrm{10}}/{J}_{\mathrm{6}}$ is higher than the
L-2007 ${J}_{\mathrm{10}}/{J}_{\mathrm{6}}$. However, this is likely due to the influence of other
sources, such as traffic emissions, on formation rates rather than due to
NPF. For a large fraction of the time, the observed median and the L-2007
${J}_{\mathrm{10}}/{J}_{\mathrm{6}}$ agree relatively well.

### 3.4.2 Survival probabilities and particle growth and losses

Figures 7,
8, and
9 show the observed survival
probabilities ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$, ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$, and ${J}_{\mathrm{10}}/{J}_{\mathrm{6}}$ in Beijing,
China, as a function of CS $/$ GR. The survival probabilities from the cluster
population simulations for both Cases 1 and 2, corresponding to the cases
with no collisions between the clusters and including them, are also shown
for comparison. Finally, the L-2007 survival probabilities are shown.
Figures 7,
8, and
9 also show *J*, CS, and GR corresponding to
the observed survival probabilities. Note that while the CS for ACDC Case 2
represents the background CoagS, and neglects the losses due to coagulation
between the clusters, we assume that this does not affect the comparability
to the observed survival probabilities since CS in Beijing is mostly
governed by accumulation mode particles (Cai et al., 2017b).

A majority of the observed ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ are larger than predicted by the L-2007 and the ACDC Case 1 and Case 2 (Fig. 7). When the ratio CS $/$ GR ≈10, the median of the observed ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}\approx \mathrm{0.2}$, while the ACDC Case 1 and the L-2007 ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}\approx \mathrm{0.1}$ and the ACDC Case 2 ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}\approx \mathrm{0.03}$. When CS $/$ GR ≈20, the corresponding median observed ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}\approx \mathrm{0.2}$. At a similar CS $/$ GR, ACDC Case 1 and L-2007 predict that ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}\approx \mathrm{0.01}$, and the ACDC Case 2 ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}\approx \mathrm{0.005}$. At CS $/$ GR ≥20, the differences in ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ between the observations and theory are considerably larger than at lower CS $/$ GR, and a much higher fraction of the particles appears to grow from 1.5 to 3 nm than what L-2007 or the cluster population simulations predict.

The ACDC Case 1 and the L-2007${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ have a strong exponential dependency on CS $/$ GR, which is not apparent in the observed ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$. While ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ appears to be on average lower when CS $/$ GR is higher, the difference is much less than expected. The same is true for ACDC Case 2 ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$. It might be possible that at sufficiently low background CS, the effect of collisions with other small particles considerably decreases the atmospheric ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$, explaining the weak dependency of observed survival probability on CS. However, this would mean that without these additional losses due to coagulation with other small particles, the observed ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ would be even higher at low CS $/$ GR, further increasing the disagreement between the theoretical predictions and atmospheric observations. In addition, Xiao et al. (2021) showed that the van der Waals attraction force, which we have not taken into account, enhances GR and leads to a weaker dependency of survival probability on CS, which could in part explain the weak dependency of the observed ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ on CS.

Unlike for ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$, the observed ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$ is on average lower
than the L-2007 and the ACDC Case 1 ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$ (Fig. 8). When CS $/$ GR ≈5, L-2007 and
the ACDC Case 1 ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}\approx \mathrm{0.5}$, while most of the observed
${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$ are lower than that, and the median of the observed
${J}_{\mathrm{6}}/{J}_{\mathrm{3}}\approx \mathrm{0.2}$. The observed ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$, like
${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$, does not show a strong dependency on the ratio CS $/$ GR. The
ACDC Case 2 values of ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$ are smaller than most of the observed
${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$; however, the difference between the observed median and the
ACDC Case 2 ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$ is mostly within a factor of 2. In addition, the
dependency on CS $/$ GR is similar both for the observations and for the ACDC Case 2
${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$ within the range of relevant values of CS $/$ GR. Thus, the
observed ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$ could be explained quite well by assuming high rates
of collisions between sub-10 nm particles when CS $/$ GR is relatively low. The
median values of the observed ${J}_{\mathrm{10}}/{J}_{\mathrm{6}}$ are relatively close to the
values of${J}_{\mathrm{10}}/{J}_{\mathrm{6}}$ based on ACDC Case 1, Case 2, and L-2007 (Fig. 9). However, the observed
${J}_{\mathrm{10}}/{J}_{\mathrm{6}}$ varies significantly and a notable fraction of the values are above
unity. At this size range, particle emissions from traffic can considerably
affect the observed *J*, thereby influencing ${J}_{\mathrm{10}}/{J}_{\mathrm{6}}$, as indicated by size-resolved particle number emissions determined for the same
measurement station (Kontkanen et al., 2020). We assume that most of the
variability in the observed ${J}_{\mathrm{10}}/{J}_{\mathrm{6}}$ can be explained by some
combination of emissions of sub-10 nm particles and measurement inaccuracies
and thus our analysis and discussion focus more on ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$ and ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$.

Table 2 shows the root mean square logarithmic error and absolute mean error of the observed survival probabilities to the L-2007 survival probabilities. L-2007 describes the observed ${J}_{\mathrm{10}}/{J}_{\mathrm{6}}$ best while the disagreement is highest for ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$. It is also clear that L-2007 is closer in describing the observed ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ when CS $/$ GR is 20 or lower. It is notable that when CS $/$ GR is 20 or lower, ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ is even slightly better predicted by theory than ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$. This is in agreement with Fig. 6, which showed a relatively good agreement between the median NPF day ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ and L-2007.

If we investigate the observed ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ with CS $/$ GR >20,
which diverges from the predictions the most, we can see that the
corresponding values of CS are quite high, while GR values are relatively small
(Fig. 7). In addition, the formation
rates, especially *J*_{1.5}, tend to be high. However, ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ with
the highest disagreements between the observations and the predictions are
mainly characterized by high values of CS. These results indicate that a
potential reason for the disagreement between the observed and the predicted
${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ is due to CS not corresponding to the actual coagulation
scavenging rates during these events. Another explanations for the
disagreement could be considerable underestimation of sub-3 nm GR in high CS
conditions or inhomogeneities in the particle formation. These will be
discussed further in Sect. 3.4.3.

From Fig. 8 we can see that the disagreement between the L-2007, ACDC Case
1, and the observed ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$ is largest when CS is low and the
disagreement does not seem to vary strongly depending on the value of GR.
This supports the possibility of coagulation between sub-10 nm particles at
low CS having a considerable effect on the survivability of particles
between 3 and 6 nm. However, current research does not support
self-coagulation having such a large contribution to the observed values of
survival probabilities since the required concentrations of sub-10 nm particles
are high (Anttila et al., 2010; Yue et al., 2010). In our simulations, the
concentrations of sub-3 nm particles need to exceed 3×10^{6} cm^{−3} before the values of ACDC Case 2 survival probabilities are
considerably lower than the values of ACDC Case 1 and the L-2007 survival
probabilities due to cluster–cluster collisions. Based on previous studies,
the concentrations of cluster and nucleation mode particles in Beijing
during NPF event days are an order of magnitude lower than that (Zhou et
al., 2020; Cai et al., 2021b), although the concentrations contain high
uncertainties (Kangasluoma et al., 2020). In addition, if self-coagulation
of sub-10 nm particles does lower atmospheric survival probabilities, we
would assume it to also be evident at smaller sizes. Despite this, it is
still possible that self-coagulation does influence the observed survival
probabilities to some extent, and more research is needed to quantify the
effect of population dynamics such as cluster–cluster collisions on survival
probabilities of sub-10 nm particles.

### 3.4.3 Effect of uncertainties and assumptions on survival probabilities

*J*, GR, and CS are determined based on measured particle number size
distributions. There can be considerable uncertainties in the measured
number size distributions, especially at the sub-10 nm size range
(Wiedensohler et al., 2012; Kangasluoma et al., 2020). CS in Beijing is
dominated by accumulation mode particles and the uncertainty of the measured
particle number size distribution in this size range is estimated to be
±10 % (Wiedensohler et al., 2012); thus we assume that the
uncertainty in CS is similar. The uncertainty in the measured particle
number size distributions of sub-10 nm particles is significantly higher and
has been estimated to be ±50 %–70 % (Kangasluoma et al., 2020). As
*J* in Beijing is mainly contributed by the concentration of new particles and
CS (Cai and Jiang, 2017), we assume the uncertainty in *J* to be in the same
range. The uncertainty in GR is also high, and we estimate that it can be up
to ±100 %. Here, the influence of uncertainties in *J*, CS, or GR on
survival probabilities and their comparison to theoretical predictions are
considered. First, we discuss the uncertainties in the formation rates, and
their effect on the observed survival probabilities. Then we discuss the
uncertainties in CS and GR, and their effect on the comparison of the
observed survival probabilities with theoretical survival probability. Figures 10 and
11 show the effect of inaccurate CS and
GR, respectively, on the L-2007 survival probabilities.

We estimate, based on Kangasluoma et al. (2020), that the uncertainties in *J* are
approximately ±50 %–70 %, increasing with a decreasing particle
diameter. At low CS $/$ GR, a majority of the observed ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ could thus
be explained by uncertainties in *J*. The observed ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ being on
average larger than L-2007 predictions could be due to larger systematic
uncertainties in *J*_{1.5} compared to *J*_{3}. However, when CS $/$ GR is
larger, the observed ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ are up to 2 magnitudes of order higher
than the L-2007 predictions, and uncertainties in the observed values of *J*
cannot explain such a discrepancy. Most of the observed values of
${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$ are approximately 2–4 times lower than the L-2007
predictions, so that while uncertainties in *J* surely contribute to the
variance of the observed ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$ and thus have a potentially
considerable contribution to the observed discrepancy, they are unlikely to
be the only explanation for the latter.

Based on Fig. 10, for ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ at CS $/$ GR ≥20 CS must be between 50 % and 75 % lower than assumed to explain the discrepancy between the observed ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ and L-2007 ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$. A large fraction of the observed ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ at CS $/$ GR <20 would also require CS to be less than half of the assumed CS to be explained by inaccuracy of CS. Most of the observed ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ at CS $/$ GR <20 can be explained if the assumed CS is between 50 % lower and 50 % higher than the true CS. Assuming that the contribution of self-coagulation is minor, to explain most of the observed ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$ by inaccuracy of CS, CS must be more than 100 % higher. Thus, we argue that the uncertainty of CS due to the uncertainties in the measured particle concentrations cannot solely explain the observed ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ or ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$.

Figure 11 considers the uncertainty of GR instead of the uncertainty of CS. Most of the observed ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ at CS $/$ GR <20 can be explained by GR ±50 %–75 % × GR. However, to explain most of the observed ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ at CS $/$ GR ≥20, and a large fraction at CS $/$ GR <20, GR would have to be higher by at least 100 % and in some cases up to 300\%̇. Thus, while most of the observed ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ at CS $/$ GR <20 can be explained by the uncertainty of GR if GR ±100 % × GR, a much higher uncertainty is needed to explain most of the observed ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ at CS $/$ GR ≥20. To explain most of the ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$ GR would have to be overestimated by 25 %–75 %. Thus, it is possible that the disagreement between the ACDC Case 1, the L-2007, and the observed ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$ is due to the estimated uncertainty of GR. However, due to the higher uncertainty required to explain the observed ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$, and the opposite direction of the required inaccuracies, it appears less likely that the uncertainty of GR could also explain, at least fully, the disagreements between the theoretical and the observed ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$.

CoagS could be overestimated or underestimated if some assumptions made when determining it are inaccurate. We have neglected the enhancement of coagulation due to van der Waals forces, which can result in underestimating CoagS. The van der Waals enhancement factor of CoagS is expected to lie between 1.0 and 2.0 (Kerminen, 1994). While this alone cannot explain the majority of the differences in ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$ between observations and predictions (Fig. 10), it could be a partial reason for the observed ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$ being lower than those predicted based on ACDC Case 1 and L-2007. In addition, all the collisions between new particles and the background particles are assumed to result in coagulation. However, it is possible that only a fraction of all the collisions lead to coagulation due to, e.g., chemical properties of the particles. This is analogous to the effectiveness of CS in removing condensable vapors such as sulfuric acid (Tuovinen et al., 2021, 2020), which has been shown to be higher for ammonium-nitrate-rich background particles (Du et al., 2022). If the effectiveness of CoagS for particles between 1.5 and 3 nm varies between 0.25 and 1.0, almost all of the observed ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ that are larger than predicted based on the simulations or L-2007 can be explained. However, for ineffective CoagS to explain the observed ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$, the effectiveness of CoagS would have to be strongly size dependent since the observed values of ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$ and ${J}_{\mathrm{10}}/{J}_{\mathrm{6}}$ do not support CoagS being lower than assumed between 3 and 10 nm.

In the atmosphere, conditions are constantly changing. We have assumed that
GR is constant throughout the event, which can result in inaccuracies in
both CS $/$ GR and the survival probabilities themselves. The effects of this
assumption on our results are less than straightforward to evaluate.
However, in Fig. 7, for example, the
highest disagreements between the predicted and the observed
${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ are characterized by high CS and thus while the
time dependency of GR might contribute to the disagreements between
predictions and observations in some way, it cannot explain the large
disagreements for ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ at CS $/$ GR ≥20. The particle number
size distribution of background particles is also constantly changing and
while we have accounted for the time dependency of CS we have not considered
the time dependency of the parameter *m* but have assumed a constant *m* of −1.6. We
expect the uncertainty from this assumption to be relatively small (see
Fig. 4) and the effect on our main
results to be minor.

While NPF events may take place regionally, the air masses where NPF occurs
are not completely homogeneous, especially in urban areas with a variety of
local emission sources. Because of this the values of CS, GR, and *J* can have
spatial variations. For example, GR could on a local scale be higher or
lower than what we determine based on the measured particle number size
distributions. This can result in locally higher survival rates of sub-10 nm
particles, which could affect the observed survival probabilities due to the
increased total concentrations of new particles. Increased GR on a local
scale could be due to, e.g., larger concentrations of precursor vapors owing to the proximity to emission sources. Wang et al. (2020) and Marten et al. (2022)
showed that new particles can grow very rapidly despite high CS and have
survival probabilities close to unity in the presence of gas-phase nitric
acid and ammonia under controlled laboratory conditions, which could be
relevant for inhomogeneous urban environments with local emission sources
such as traffic. Thus, while brief but rapid growth of new particles on a
small local scale might have no effect on the observed GR at the measurement
location, it could result in significantly higher values of observed survival
probabilities causing apparent discrepancy between the observations and
theory.

Another factor relevant in urban environments to be considered is the effect of primary particle emissions from traffic on the measured particle number size distributions. Traffic emissions have been shown to significantly contribute to number concentrations of particles as small as below 3 nm in diameter (Rönkkö et al., 2017). However, Deng et al. (2022) showed that the influence of traffic emissions on the concentration of sub-3 nm particles at the observation site is negligible compared to the influence of NPF. Thus, the effect of primary particle and cluster emissions from traffic on ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ is likely minor.

We note that it should be considered when interpreting our results that we are only able to observe NPF events when both the survival probabilities and the formation rates themselves are sufficiently high. Thus, it is likely that our results are biased toward higher survival probability values, which could make the discrepancy between the observed and the predicted ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ appear more significant. If we were also able to accurately describe the growth and formation rates of sub-10 nm particles outside these events, we would have a more complete picture of survival probabilities and their dependency on coagulation scavenging and particle growth rates in urban Beijing.

We compared cluster or aerosol particle survival probabilities from 1.5 to 3 nm (${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$), from 3 to 6 nm (${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$), and from 6 to 10 nm (${J}_{\mathrm{10}}/{J}_{\mathrm{6}}$) between predictions based on analytical formulae, cluster population simulations using Atmospheric Cluster Dynamics Code (ACDC), and observations in Beijing, China, and discussed possible reasons for the corresponding differences. The survival probabilities based on theory and the cluster population simulations agree relatively well for all of the three size ranges if no cluster–cluster collisions occur in the simulations or if their contribution to growth and losses of clusters are negligible. However, if CS is low, the inclusion of cluster–cluster collisions in the cluster population simulations results in significantly lower survival probabilities for all the investigated size ranges, and in a weaker dependency of the survival probability on CS due to the increased loss rate of clusters and particles.

A majority of the observed values of ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ are higher than those obtained from the cluster population simulations or based on analytical formulae, and the largest discrepancies were observed at high values of CS. At low CS $/$ GR a majority of the observed values of ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ can be explained if uncertainties in CS $/$ GR reach approximately ±75 %, a reasonable estimate for an error in atmospheric CS $/$ GR. However, at higher CS $/$ GR the value of CS needs to be lower by more than 50 % or the value of GR needs to be higher by more than 100 % to explain the observed values of ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$. Unlike for ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$, a majority of the observed ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$ are lower than theoretical ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$ and the discrepancy is even higher than for ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ when CS is low. The disagreement between the theoretical and the observed ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$ can be explained if CS is underestimated by more than 100 % or if GR is overestimated by approximately 25 %–75 %. However, the observed ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$ and ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$ from the cluster population simulations with cluster–cluster collisions are closer, mostly within a factor of 2 from each other. The median values of the observed ${J}_{\mathrm{10}}/{J}_{\mathrm{6}}$ are relatively close to the theoretical values and to ${J}_{\mathrm{10}}/{J}_{\mathrm{6}}$ from the cluster population simulations, both with or without cluster–cluster collisions. However, the variance of these values is high, which we attribute both to measurement uncertainties and to the influence of emissions of sub-10 nm particles to the observed formation rates. Thus, only the survival probability from 1.5 to 3 nm appears to be on average higher than predicted under high CS, while the survival probabilities between larger sizes did not show a similar trend.

Based on our results it appears unlikely that the effect of cluster–cluster collisions on survival probability explains the observed discrepancies between theory and observations. However, more research is needed to quantify the role of complex dynamic interactions between sub-10 nm particles on survival probability. A reasonable overestimation within the limits of estimated uncertainties in GR can potentially explain the observed values of ${J}_{\mathrm{6}}/{J}_{\mathrm{3}}$ if the influence of cluster–cluster collisions is assumed to be negligible. While a large fraction of the observed values of ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ can be explained by the uncertainties of measured CS and GR, it seems probable that at high CS conditions, other factors also contribute to the observed survival probabilities. Possible explanations for the observed values of ${J}_{\mathrm{3}}/{J}_{\mathrm{1.5}}$ under high background particle concentrations include overestimation of CS due to ineffective coagulation scavenging of sub-3 nm particles, or strongly enhanced growth of sub-3 nm particles on a local scale, which is not visible in the observed GR at the measurement location. More research is still required to determine the mechanisms behind enhanced survival probability of sub-3 nm particles in polluted conditions.

Simulation datasets and the observed survival probabilities, along with the corresponding CS and GR, are available at https://doi.org/10.5281/zenodo.7247561 (Tuovinen, 2022).

The supplement related to this article is available online at: https://doi.org/10.5194/acp-22-15071-2022-supplement.

JK conceptualized the original idea. ST performed the analysis with help from JK, RC, VMK, and MK. CY, MK, and JJ collected the data. ST prepared and edited the manuscript with contributions from all co-authors.

At least one of the (co-)authors is a member of the editorial board of *Atmospheric Chemistry and Physics*. The peer-review process was guided by an independent editor, and the authors also have no other competing interests to declare.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We acknowledge the following projects: ACCC Flagship funded by the Academy of Finland grant number 337549, Academy professorship funded by the Academy of Finland (grant no. 302958), Academy of Finland project nos. 1325656, 311932, 316114, 332547,325647, and 332547; “Quantifying carbon sink, CarbonSink+ and their interaction with air quality” INAR project funded by Jane and Aatos Erkko Foundation, European Research Council (ERC) project ATM-GTP Contract No. 742206. Technical and scientific staff in Beijing, AHL/BUCT laboratory are acknowledged.

This research has been supported by the Academy of Finland (grant nos. 337549, 302958, 1325656, 311932, 316114, 332547, 325647, and 332547), the European Research Council, H2020 European Research Council (ATM-GTP (grant no. 742206)), and the Jane ja Aatos Erkon Säätiö (Quantifying carbon sink, CarbonSink+ and their interaction with air quality).

Open-access funding was provided by the Helsinki University Library.

This paper was edited by Markus Petters and reviewed by two anonymous referees.

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