the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Survival probability of new atmospheric particles: closure between theory and measurements from 1.4 to 100 nm

### Runlong Cai

### Chenjuan Deng

### Dominik Stolzenburg

### Chenxi Li

### Junchen Guo

### Veli-Matti Kerminen

### Jingkun Jiang

### Markku Kulmala

### Juha Kangasluoma

The survival probability of freshly nucleated particles governs the influences of new particle formation (NPF) on atmospheric environments and the climate. It characterizes the probability of a particle avoiding being scavenged by the coagulation with pre-existing particles and other scavenging processes before the particle successfully grows up to a certain diameter. Despite its importance, measuring the survival probability has been challenging, which limits the knowledge of particle survival in the atmosphere and results in large uncertainties in predicting the influences of NPF. Here we report the proper methods to retrieve particle survival probability using the measured aerosol size distributions. Using diverse aerosol size distributions from urban Beijing, the Finnish boreal forest, a chamber experiment, and aerosol kinetic simulations, we demonstrate that each method is valid for a different type of aerosol size distribution, whereas misapplying the conventional methods to banana-type NPF events may underestimate the survival probability. Using these methods, we investigate the consistency between the measured survival probability of new particles and the theoretical survival probability against coagulation scavenging predicted using the measured growth rate and coagulation sink. With case-by-case and time- and size-resolved analysis of long-term measurement data from urban Beijing, we find that although both the measured and theoretical survival probabilities are sensitive to uncertainties and variations, they are, on average, consistent with each other for new particles growing from 1.4 (the cluster size) to 100 nm.

As one of the largest sources of uncertainties in climate prediction, atmospheric aerosol plays a key role in diverse environments. New particle formation (NPF) via vapor nucleation and subsequent growth (Kulmala et al., 2004, 2013; Zhang et al., 2012) is a key phenomenon associated with the atmospheric aerosol system, contributing majorly to the number concentrations of aerosols and cloud condensation nuclei (Kuang et al., 2009; Gordon et al., 2017). To reach a relatively long atmospheric residence time and exert significant influences on atmospheric environments, freshly nucleated particles need to grow fast beyond the smallest size range (e.g., sub-10 nm), in which they are most likely to be scavenged by pre-existing aerosols (McMurry, 1983; Kulmala et al., 2001). Hence, the fraction of freshly nucleated particles that survive the scavenging after a certain growth process, characterized by the survival probability, is a decisive factor for the influence of NPF on the atmosphere.

Although the formation rate (*J*) and growth rate (GR) of new particles have
been often used to characterize NPF events (Kulmala et al., 2012;
Kerminen et al., 2018), the survival probability is an important
irreplaceable parameter in NPF studies. *J* and GR can connect the gaseous
precursors and new particles, providing information on the formation and
growth mechanisms. In terms of atmospheric influences, however, it is the
total number of new particles formed during NPF and their survival
probabilities that determine the contribution of NPF to the number of large
particles. While the total number of new particles can be readily obtained
by integrating *J* over time, predicting the survival probability requires
information on the sink of particles in addition to GR. Although several
studies computed the survival probability using the ratio of *J* at different
particle sizes, we show below that this method may not be valid for every
type of NPF event.

The knowledge of particle survival probability in the real atmosphere is rather limited despite its importance. On one hand, theoretical survival probabilities predicted using GR and the coagulation sink (CoagS) of new particles (Kerminen and Kulmala, 2002; Lehtinen et al., 2007; Pierce and Adams, 2007) have been widely used in regional and global models, bridging the gap between the size of freshly nucleated particles and the smallest size bin/mode in the model. On the other hand, however, there have been only a limited number of studies reporting the measured survival probabilities retrieved from aerosol size distributions (e.g., Weber et al., 1997; Kuang et al., 2009; Kulmala et al., 2017; Zhu et al., 2021; Sebastian et al., 2021). As summarized in the Appendix, only few studies have compared the measured and theoretical survival probabilities, and the reported results seem to indicate that the measured survival probability was sometimes higher than theoretical values. The limited information from measurements is the main obstacle for validating the modeling results and understanding particle survival. For instance, Kulmala et al. (2017) reported that for sub-3 nm particles in polluted megacities, the theoretically predicted survival probability could not explain the measured value with a deviation of several orders of magnitude, urging further investigations of the survival of freshly nucleated particles.

Retrieving the size-resolved survival probability from measured aerosol size
distributions is the first step to address the consistency problem between
measurements and theory, yet it has been challenging. The challenge mainly
comes from the difficulty to track the growth and survival of individual
particles or an aerosol population, as the measured particles are a sum of
the surviving particles and other newly formed particles. Therefore, one has
to retrieve survival probability from measurements based on other parameters
instead of the total number of growing particles. Weber et
al. (1997) used the concentration of gaseous sulfuric acid and the
theoretical survival probability to estimate the total concentration (*N*) of
particles larger than 3 nm, which essentially approximated the measured
survival probability with the ratio of *N* at different times. Kerminen
and Kulmala (2002) derived the theoretical survival probability of a
strictly monodisperse aerosol population and concluded that the survival
probability should be equal to the ratio of *J* at different sizes.
Kuang et al. (2009) derived a semi-analytical formula for $n=\mathrm{d}N/\mathrm{d}\phantom{\rule{0.125em}{0ex}}{d}_{\mathrm{p}}$ of an aerosol population with the influence of time-dependent
source and sink terms, where *n* is the aerosol size distribution on the linear
scale of particle size. The ratio of *n* at different sizes was taken as the
survival probability. However, the derivations and validations of these
formulae using *J* and *n* may need further discussion. Here we demonstrate that
each formula is valid for only a certain type of NPF events, and it may
report substantially biased survival probabilities for other events. These
potential biases demand advances in the methods to retrieve the measured
survival probability.

In this study, we report the proper formulae to retrieve the measured
survival probability of new atmospheric particles. In addition to
conventional formulae based on *J* and *n*, we propose a new formula based on
${n}_{\mathrm{log}}=\mathrm{d}N/\mathrm{d}\phantom{\rule{0.125em}{0ex}}\mathrm{log}\phantom{\rule{0.125em}{0ex}}{d}_{\mathrm{p}}$ to retrieve the measured survival probability from
a growing aerosol population (e.g., in a banana-type NPF event). The
performance of these formulae with different types of size distributions is
tested using benchmark simulations based on aerosol kinetics and
measurements from the real atmosphere, showing that these formulae are valid
for their corresponding types of aerosol size distributions.

We then use the proper formulae to investigate the consistency between the measured and theoretical survival probabilities. Using data from long-term measurements in urban Beijing, a case study measured from a Finnish boreal forest site, and a chamber experiment, we perform case-by-case and time- and size-resolved analysis of particle survival probabilities. The results show that despite the large variations and high sensitivity in the survival probability, especially for sub-5 nm particles, the measured survival probability of new particles growing from 1.4 (electrical mobility diameter) to 100 nm can on average be explained by theory.

## 2.1 Definition of survival probability

The survival probability of a growing individual particle is the probability
that the particle at an initial diameter (*d*_{p1}, nm) will grow to a
specified diameter (*d*_{p2}) before it is scavenged
(Weber et al., 1997; Pierce and Adams, 2007). For a
growing monodisperse aerosol population containing a large number of
particles in a steady environment, the survival probability is equal to the
fraction of particles successfully growing from *d*_{p1} to *d*_{p2} against
the scavenging, i.e.,

where *P* (–) is the survival probability, *N* (cm^{−3}) is the total number
concentration of particles in the monodisperse population, *t* (s) is time, and
the subscript indicates the particle size and number concentration of the
aerosol population corresponding to a certain time. Equation (1) emphasizes
that the survival probability is defined for the same aerosol population and
that the aerosol concentration is not affected by atmospheric processes of
transport, mixing, dilution, etc.

## 2.2 Theoretical survival probability

The theoretical survival probability, referred to as *P*_{theo} below, can be
predicted using time- and size-dependent scavenging losses. For freshly
nucleated atmospheric particles, scavenging losses are usually governed by
Brownian coagulation (Kulmala et al., 2001). With a
reasonable approximation that particles in the growing population share the
same time- and size-dependent coagulation sink, the theoretical probability
against coagulation scavenging can be computed by integrating CoagS as a
function of particle size and time, i.e.,

where *N* is the total concentration of particles in the growing population.
Further assuming that particles share the same time- and size-dependent GR
(nm s^{−1}) and substituting d$t=\mathrm{d}\phantom{\rule{0.125em}{0ex}}{d}_{\mathrm{p}}/$GR and $N={N}_{\mathrm{1}}\cdot {P}_{\mathrm{theo}}$ into the derivative of Eq. (2), one can obtain
$\mathrm{d}{P}_{\mathrm{theo}}/{P}_{\mathrm{theo}}=-\mathrm{d}{d}_{\mathrm{p}}\cdot \mathrm{CoagS}/\mathrm{GR}$. Integrating this differential formula yields Eq. (3), in which *P* is expressed as a function of *d*_{p},

Equation (3) can be further simplified with predetermined size dependencies of
CoagS (s^{−1}) and GR, as have been given in previous studies (Weber et
al., 1997; Kerminen and Kulmala, 2002; Lehtinen et al., 2007; Korhonen et
al., 2014). These simplified formulae can be readily applied to compute
*P*_{theo} without CoagS and GR for each *d*_{p}, yet in this study we use
Eqs. (2) and (3) for better accuracy.

## 2.3 Measured survival probability

Different from the theoretical survival probability predicted using CoagS
and GR (Eqs. 2 and 3), the measured survival probability is retrieved from
the size distribution of growing new particles. Because of the challenges in
tracking the same growing aerosol population against other freshly nucleated
or pre-existing particles, it is practically difficult to use the definition
of the survival probability in Eq. (1) for atmospheric measurements. As
previously mentioned, particle formation rate *J* (cm^{−3} s^{−1};
Kerminen and Kulmala, 2002) and linear size-scale distribution $n=\mathrm{d}N/\mathrm{d}\phantom{\rule{0.125em}{0ex}}{d}_{\mathrm{p}}$ (cm^{−3} nm^{−1}; Kuang et al., 2009) have been used
as the parameters to retrieve particle survival probability from a growing
aerosol population. In this study, however, we find that *J* and *n* are valid for
steady-state and quasi-steady-state size distributions but may not be for a
growing aerosol population, whereas the logarithmic size-scale ${n}_{\mathrm{log}}=\mathrm{d}N/\mathrm{d}\phantom{\rule{0.125em}{0ex}}{\mathrm{log}}_{\mathrm{10}}{d}_{\mathrm{p}}$ is a promising empirical parameter for a growing aerosol
population. The corresponding formulae for retrieving the measured survival
probability are given in Eqs. (4)–(6):

where subscripts 1 and 2 indicate the diameters *d*_{p1} and *d*_{p2},
respectively, at which *J*, *n*, and *n*_{log} are evaluated. For the convenience
of illustration, we refer to the survival probabilities retrieved using *J*,
*n*, and *n*_{log} as *P*_{J}, *P*_{n}, and $P{}_{{n}_{\mathrm{log}}}$, respectively.

## 2.4 Theoretical analysis of the measured survival probability

Here we present a theoretical analysis of the validity of Eqs. (4)–(6) using two ideal types of aerosol size distributions. First, for a growing aerosol population, which is assumed to follow a lognormal distribution, the survival probability can be expressed as Eq. (7) according to the definition in Eq. (1).

where *n*_{log} is the value at the distribution peak, and *σ*_{g} is
the geometric standard deviation of aerosol size distribution.

For a growing aerosol population in the atmosphere, it is an empirical
conclusion that *σ*_{g} usually stays at a relatively constant
level (e.g., Hussein et al., 2004). Figure 1 shows an
example of banana-type NPF events with a clear growth pattern of new
particles from 5 to 60 nm. Although there were minor variations in
*σ*_{g}, the maximum *n*_{log} was relatively constant after
normalizing the change in *N*. This indicates that neglecting the ln(${\mathit{\sigma}}_{\mathrm{g},\mathrm{2}})/$ln(*σ*_{g,1}) term in Eq. (7) for these events would
introduce only a minor uncertainty, and hence ${P}_{n{}_{\mathrm{log}}}$ in Eq. (6) could
provide a good estimate of the survival probability of particles in this
growing population. In contrast, *n* was significantly broadened as *d*_{p}
increased, and the maximum *n* decreased due to this broadening. This indicates
that *P*_{n} in Eq. (5) would underestimate the survival probability of the
growing aerosol population. Since $J=n\cdot $ GR, *P*_{J} in Eq. (4) would
also underestimate the survival probability.

It is worth clarifying that *σ*_{g} may have a strong size
dependency during some growth processes. For instance, if particle growth is
only driven by the condensation of non-volatile vapors, *σ*_{g}
tends to decrease as particles grow in size (see the Appendix). For those
kinds of situations, Eq. (7) should be used instead of Eq. (6) for better
accuracy.

We then consider another type of ideal aerosol size distribution, for which
the *n* of freshly nucleated particles is at a steady state. The steady state
refers to a condition under which the time derivative of *n* is negligible, i.e.,

where the ∂*J*(*d*_{p})
term characterizes the flux of growing particles
through *d*_{p}.

The formulae for the measured survival probability of particles with this
steady-state distribution can be theoretically derived. Substituting $n=J/$GR into Eq. (8) and integrating from *d*_{p1} to *d*_{p2} yields the
relationship between *J* and CoagS/GR,

Comparing Eqs. (3) and (9), one can readily conclude that ${J}_{\mathrm{2}}/{J}_{\mathrm{1}}$ is equal to the survival probability of particles with a steady-state size distribution. With a size-independent GR, ${n}_{\mathrm{2}}/{n}_{\mathrm{1}}$ is also equal to the survival probability, i.e.,

It is worth emphasizing that although Eqs. (10) and (11) look very similar to Eqs. (4) and (5), there is a conceptual difference between them. This difference lies in the view of the two ideal aerosol size distributions. For the growing aerosol population, we track the time evolution of the aerosol size distribution from a Lagrangian point of view in the particle size space (Eqs. 4–6). For the steady-state aerosol size distribution, the measured size distribution is the sum of particles formed at different times. Instead of tracking the temporal evolution, we focus on the particle growth fluxes through certain size bins at the same moment and then derive the relationship between the measured size distribution and the survival probability (Eqs. 10 and 11).

Summarizing the analysis above, ${P}_{{n}_{\mathrm{log}}}$ is expected to be valid for a
growing aerosol population, whereas *P*_{J} and *P*_{n} are expected to be
valid for a steady-state distribution.

## 3.1 Simulation

A sectional model based on aerosol kinetics was used as a benchmark to generate aerosol size distributions and test the formulae for survival probability computation. The evolving aerosol size distribution was simulated by numerically solving aerosol population balance equations, which account for new particle formation, particle condensational growth, the coagulation sink of particles, and self-coagulation of new particles. Detailed information on this model has been described previously (Li and Cai, 2020). We validated the accuracy of the sectional model using a discrete model, ensuring that numerical diffusion did not affect the conclusions based on simulation results. For the convenience of discussion, we used time-independent external coagulation sinks for particles in these simulations. A size-dependent growth enhancement factor for particle growth (Kuang et al., 2010) was used as an input parameter to generate size-dependent growth rates. The simulation conditions are summarized in Table 1.

## 3.2 Measurements

We investigated the survival probability of new particles and the validity
of different methods for survival probability computation using data
measured from urban Beijing and a Finnish boreal forest site. The long-term
NPF data for urban Beijing was measured at the BUCT (Beijing University of Chemical Technology) site (Liu et al.,
2020), which is located on the west campus of Beijing University of Chemical
Technology and close to the west 3rd Ring Road of Beijing. Despite the
high coagulation sink of new particles in urban Beijing (Cai et al.,
2017b), intensive NPF events have been frequently observed. We used the data
measured from 16 January to 26 December 2018 in Beijing to represent
NPF events in polluted megacities and analyzed a total of 65 NPF event days
with clear patterns of new particle formation and growth. The NPF event
measured on 11 April 2020 from a relatively clean environment at the
SMEAR II station at Hyytiälä, Finland (Hari and Kulmala, 2005)
was analyzed as a case study. We also analyzed an NPF experiment measured in
the Cosmics Leaving OUtdoor Droplets (CLOUD) chamber at the European Center
for Nuclear Research (CERN; Kirkby et al., 2011; Duplissy et al., 2016).
In that experiment, gaseous precursor (*α*-pinene, isoprene, and
ozone) concentrations were kept constant, and NPF was initiated by a sudden
increase in ion concentrations in the chamber (Heinritzi et al., 2020),
yielding a steady-state aerosol size distribution during NPF.

The aerosol number size distributions in the Beijing atmosphere were measured using a diethylene glycol scanning mobility particle spectrometer (DEG-SMPS; Jiang et al., 2011) and a particle size distribution system (PSD; Liu et al., 2016). The DEG-SMPS covered the size range of 1–6.5 nm (electrical mobility diameter, same below), and it was deployed with a core sampling device (Fu et al., 2019) and a miniature cylindrical differential mobility analyzer (Cai et al., 2017a) to improve the sampling and classification of particles in this size range. The PSD was used to measure particles with diameters ranging from 3 nm to 10 µm. The aerosol size distributions at Hyytiälä were measured using a differential mobility analyzer train (DMA train; Stolzenburg et al., 2017) and a twin differential mobility particle spectrometer (Aalto et al., 2001), covering the diameter ranges of 1.8–8 and 3–1000 nm, respectively. At CLOUD, the aerosol size distributions were measured by the DMA train (1.8–8 nm) and a TSI nano-scanning mobility particle sizer (nano-SMPS; Model 3982, 2–64 nm). All these instruments obtained the size information of aerosols based on the electrical mobility classification, which could provide a relatively good sizing accuracy, especially for freshly nucleated particles.

## 3.3 Data analysis

The formation rate *J*, CoagS, and GR along the growth trajectories of new
particles were determined to compute the survival probability of new atmospheric particles. For simulated NPF events, the growth trajectory was obtained
using a monodisperse aerosol model. For measured NPF events, the growth
trajectory is approximated using the evolution of particle mode diameters or
the maximum concentration method, though particle growth does not exactly
follow the increasing diameters due to the influences of coagulation
(Stolzenburg et al., 2005; Leppä et al., 2011). *J* was retrieved using
a population balance method with improved accuracy for NPF in polluted
environments (Cai et al., 2017a). Time- and size-dependent CoagS
was computed using the measured particle size distribution
(Kulmala et al., 2001).

We tested the performance of different methods in different environments in
order to minimize the uncertainties in the retrieved GR. For urban Beijing,
we found a systematic difference between the GR estimated using the
appearance time method (Lehtipalo et al., 2014; Cai et al., 2021) and the
mode-fitting method (Kulmala et al., 2012; Deng et al., 2020) for sub-5 nm particles, which has been reported previously (Qiao et al., 2021; Deng
et al., 2021). Such a difference is likely due to the influences of the
continuous formation of freshly nucleated particles and primary emissions of
particles. The mode-fitting method tracks the growth of the peak diameter of
a new particle mode by fitting lognormal distributions to the measured
*n*_{log}. However, the fitted mode is a sum of the growing particles and
freshly nucleated particles, and hence the apparent growth of the fitted mode
is slower than the growing mode. Consequently, the mode-fitting method tends
to underestimate the GR for sub-5 nm particles (Cai et al.,
2022). Hence, we used the 50 % appearance time method to calculate the GR
for sub-5 nm particles and applied a GR-based correction when computing
*P*_{theo}, which tracks the time that particle concentration for each size
bin reached 50 % of the daily maximum. For Hyytiälä, the GR was
retrieved using the maximum concentration method (Kulmala et al., 2012),
which finds the time corresponding to the maximum particle concentration in
each size bin, and the concentration was smoothed with a span of 12 min. The
GR retrieved using the maximum concentration method was consistent with that
retrieved using the 50 % appearance time method. We did not use the
appearance method directly because it does not track the growth trajectory
of new particles.

The measured survival probabilities of particles were computed using Eqs. (4)–(6). According to our simulation results, we determined *n* and *n*_{log} for
each particle size in Eqs. (5) and (6) along smoothed growth trajectories and the
*J* in Eq. (4) as the maximum *J* during an NPF event as a function of particle
size.

The *P*_{theo} for new atmospheric particles was computed using Eq. (2), with
the CoagS determined along the mode diameters and the concentration of new
particles numerically solved by iteration. This approach is equivalent to
using Eq. (3) with time- and size-dependent CoagS and GR.

To compare the theoretical and measured survival probabilities from long-term measurements in urban Beijing, we first computed their values for each NPF event. Considering the fact that the validity of the formulae to retrieve the measured survival probability in Eqs. (4)–(6) is based on the homogeneity of the system, we compared the medians of the theoretical and measured values. The medians were first computed for each size bin, and the overall median survival probabilities were then reconstructed by integrating the size-resolved values from 1.4 to 100 nm. In this way, we minimized the influences of atmospheric variations and preserved the non-linearity of the survival probability as a function of GR and CoagS.

## 4.1 Validity of P_{J}, P_{n}, and P${}_{{n}_{\mathrm{log}}}$ for different types of NPF events

We first test the methods to retrieve the survival probability of particles
in a growing aerosol population against coagulation scavenging. Vapor
concentration is constant during particle growth, and the growth rate for
10–50 nm particles shows only a weak size dependency. For a simulated system
without particle sources from nucleation, primary emissions, etc. (Fig. 2a),
the “true” survival probability can be retrieved with *N* using the
definition in Eq. (1) since all the particles are from the same population.
*P*_{theo} can be predicted using Eqs. (2) and (3) with an approximation that the
particles in this growing population share the same size-dependent CoagS and
GR. As shown in Fig. 2b, the accuracy of *P*_{theo} is evidenced by its
consistency with the “true” values. Accordingly, we use *P*_{theo} below as
a benchmark for the survival probability retrieved from every simulation
result.

The measured survival probability of particles in a growing population can
be estimated using ${P}_{{n}_{\mathrm{log}}}$. Figure 2b shows that for a growing
aerosol population with a relatively constant *σ*_{g}, the decrease
in *n*_{log} along the growth trajectory is consistent with the decreasing
theoretical survival probability, as the decrease in the maximum *n*_{log} is
mainly caused by the coagulation scavenging. In contrast, *P*_{J} and
*P*_{n} underestimate the survival probability because they neglect the
broadening of particle size distribution in the linear size scale. These
underestimations can be estimated according to ${n}_{\mathrm{log}}=n\cdot \mathrm{ln}\mathrm{10}\cdot {d}_{\mathrm{p}}$. For instance, for a population of particles growing
from 2 to 20 nm, *P*_{J} and *P*_{n} are expected to report survival
probabilities that are approximately 1 order of magnitude (0.1=2 nm$/$20 nm) lower than ${P}_{n{}_{\mathrm{log}}}$.

The validity of ${P}_{n{}_{\mathrm{log}}}$ for an ideal growing aerosol population can
be generalized to the growth of new atmospheric particles, e.g., in a
banana-type NPF event. Figure 2c shows a typical banana-type NPF event
measured at Hyytiälä, with the formation of new particles around
noon and a clean pattern of subsequent particle growth. The growth
trajectory of new particles is indicated using the maximum concentration of
particles measured in each size bin, with the maximum concentration mainly
contributed by particles formed with the maximum formation rate (at noon;
Kulmala et al., 2012). For this NPF event, ${P}_{n{}_{\mathrm{log}}}$ is consistent
with the *P*_{theo} predicted using the measured CoagS and GR, whereas
*P*_{J} and *P*_{n} substantially underestimate particle survival probability
(Fig. 2d).

We then test the methods to retrieve the survival probability of particles from steady-state distributions. As shown in Fig. 3a, new particles in the simulated system are generated with a constant formation rate, and they grow with a time-independent GR. The size distribution of freshly nucleated particles reaches a steady state shortly after the initial state, though the whole system is at a pseudo-steady state with net production of large particles. Due to the continuous particle formation, it is difficult to apply the definition of survival probability in Eq. (1) to the simulated NPF. Misapplying Eq. (1) by taking all the measured particles as surviving particles would result in survival probability values larger than unity (Fig. 3b).

*P*_{J} and *P*_{n} are valid for particles with steady-state concentrations.
For the simulated NPF in Fig. 3a, we compute *P*_{J} and *P*_{n} using the
measured aerosol size distribution at the end of the simulation (*t*=24 h)
instead of following a growth trajectory (see Eqs. 10 and 11). Consistent
with the derivations in theory, *P*_{J} can reproduce *P*_{theo} well with
negligible uncertainties, and *P*_{n} is also a good estimate with minor
uncertainties originating from the size-dependent GR. Accordingly, *P*_{J}
and *P*_{n} in Eqs. (10) and (11) can be used for NPF with constant formation and
growth rates, as well as freshly nucleated particles whose concentration is
at a steady state.

The validity of *P*_{J} and *P*_{n} for steady-state size distributions is
verified using an NPF event measured in the CLOUD chamber. As shown in Fig. 3c, with a constant formation rate of freshly nucleated particles, the
measured size distribution of sub-20 nm particles reached a steady state
after an elapsed time of ∼3 h. We took the size distribution
measured at 3.5 h and computed *P*_{J} and *P*_{n} using Eqs. (10) and (11).
${P}_{n{}_{\mathrm{log}}}$ was also computed using the same size distribution.
*P*_{theo} was computed using the measured GR and the total sink of
particles. Instead of the CoagS, the sink of new particles in the CLOUD
chamber was governed by dilution and size-dependent wall losses
(Stolzenburg et al., 2020). For this NPF event, *P*_{J} and *P*_{n} are
consistent with *P*_{theo}, while ${P}_{n{}_{\mathrm{log}}}$ overestimates the survival
probability as it is expected (Fig. 3d).

Atmospheric NPF events are usually driven by precursors with varying
concentrations. The growth of freshly nucleated particles usually takes
hours, whereas the particle formation rate usually varies significantly
during such a long period, indicating that the steady-state assumption may
not be valid for all particle sizes. As shown in Fig. 4a, freshly nucleated
particles are formed between 0–5 h in a simulated system, and the nucleation
rate as a function of time peaks at 2.5 h. These nucleated particles grow
with a time- and size-independent GR. Figure 4b shows the *P*_{J} in Eq. (10)
with the steady-state assumption evaluated at *t*=2, 3, and 4 h. Due to the
varying GR, *P*_{theo} also varies with time. Comparing *P*_{J} and the
corresponding *P*_{theo} shows that the steady-state assumption is only valid
for sub-5 nm particles in the simulated NPF, and the size range for this
validity is even narrower at the beginning of NPF (e.g., *t*=2 h).

We find that after considering the evolution of particle size distribution
along the growth trajectory, *P*_{J} and *P*_{n} can provide good estimates of
particle survival probability for the quasi-steady-state size distributions
in Fig. 4a. Different from the results in Fig. 4b, we compute the *P*_{J} and
*P*_{n} along the growth trajectory (using Eqs. 4 and 5, respectively) to
account for the variation in *J* and *n*. As shown in Fig. 4c, the *P*_{J} and
*P*_{n} along the growth trajectory are consistent with *P*_{theo} for sub-25 nm particles. When the quasi-steady state is no longer valid as particles on
the trajectory grow above 25 nm, the growth of particles is similar to the
case in Fig. 2a, and hence ${P}_{n{}_{\mathrm{log}}}$ follows *P*_{theo} better than *P*_{J}
and *P*_{n} in Fig. 4d.

The above analysis based on the simulated NPF is applicable to atmospheric
NPF events measured in urban Beijing. As the example shown in Fig. 5a,
intensive NPF was measured between 09:00 and 15:00 LT (UTC+8) on an NPF day, forming a
high concentration of new particles that grew subsequently to
∼70 nm within the same day. A growth trajectory of new
particles was obtained by tracking the fitted mode diameter of new
particles. *P*_{theo} was predicted using the measured CoagS and GR, with the
GR for sub-5 nm particles corrected with the appearance time method (see
“Methods” section). With the high formation rate and high CoagS, it could be
approximated that the size distribution of sub-10 nm particles was at a
quasi-steady state. Accordingly, the *P*_{J} and *P*_{n} retrieved along the
growth trajectory are consistent with *P*_{theo} in the sub-10 nm size range
(Fig. 5d). For the growth of particles from 25 nm to larger sizes,
*P*_{theo} is more consistent with ${P}_{n{}_{\mathrm{log}}}$ than *P*_{n} since the
measured growing size distribution was mainly contributed by the same
population of new particles formed before 18:00 LT (UTC+8).

To summarize, the survival probability of particles in a growing population
can be retrieved using the measured *n*_{log} (Eq. 6), and the evolution of
*σ*_{g} can be accounted for using Eq. (7). For quasi-steady-state
size distributions with the continuous formation of new particles, the
survival probability can be retrieved using *J* and *n*. Some atmospheric NPF
events may be composed of quasi-steady-state size distributions with
continuous particle formation and subsequent growth of the new particle
population; hence the survival probability can be estimated using a
combination of *P*_{n} or *P*_{J} for small particles and ${P}_{n{}_{\mathrm{log}}}$ for
large particles.

## 4.2 Consistency between measured and theoretical survival probabilities based on long-term measurements in Beijing

With the formulae to retrieve particle survival probability as presented
above, we investigate particle survival based on long-term measurements in
urban Beijing. The measured survival probabilities of particles growing from
1.4 to 25 nm and from 25 to 100 nm were approximated by *P*_{n} and
${P}_{n{}_{\mathrm{log}}}$, respectively. *P*_{J} of sub-10 nm particles was also
computed. We combine different formulae because, for most NPF events in
urban Beijing, the size distribution of freshly nucleated particles is at a
quasi-steady state, whereas the formation rate of freshly nucleated particles
is negligible when the growing particles are larger than 25 nm.

With case-by-case and time- and size-resolved analysis, we find that the
measured survival probability is on average consistent with the theoretical
prediction. Figure 6 shows the reconstructed median survival probabilities,
for which the measured values characterized by *P*_{n} and ${P}_{n{}_{\mathrm{log}}}$ are
consistent with *P*_{theo} for the whole size range from 1.4 to 100 nm.
This consistency is also supported by the measured *P*_{J} of sub-10 nm
particles, which is comparable to *P*_{theo}.

How new particles could survive against the
high CoagS in polluted megacities (e.g.,
Kulmala et al., 2017) has been a puzzle, as the CoagS therein was thought to be so high that
it would scavenge nearly all the freshly nucleated particles. In order to
explain the observed frequent NPF events, the CoagS has been hypothesized to
be ineffective such that the sticking probability between a new particle and
a large particle (i.e., scavenger) would be significantly below unity
(Kulmala et al., 2017). Alternatively, it has been
hypothesized that the GR of freshly nucleated particles could be extremely
high (>100 nm h^{−1}) such that these particles could grow
rapidly through the smallest sizes and become less vulnerable to coagulation
scavenging (Wang et al., 2020). However, here we address this puzzle by
showing the good consistency between the theoretical and measured median
survival probabilities of new particles from the cluster size
(∼1.4 nm) to the cloud condensation nucleus size
(∼100 nm). Note that *P*_{J} is used for sub-10 nm particles,
which is consistent with the method in previous studies. This closure
evidences that assuming an effective CoagS, the survival of new particles in
urban Beijing can on average be explained by theory with a GR similar to
that in clean environments (<10 nm h^{−1}). This consistency is
also consistent with our previous finding that NPF in urban Beijing tends to
occur on days with low CoagS (Cai et al., 2017b), as particle survival
probability on these days is expected to be orders of magnitude higher than
the probability on haze days. For instance, Kulmala et al. (2022) have
shown that the survival probability decreases sharply as the CoagS increases
above 0.01 s^{−1}. However, NPF events can occasionally be observed under
high CoagS, and there are deviations between the *P*_{J} and *P*_{theo} of
sub-3 nm particles during these events (Tuovinen et al., 2022).

## 4.3 Uncertainties in particle survival probabilities

Despite the consistency in terms of median values, large deviations are
sometimes observed between the measured and theoretical survival
probabilities (Fig. 6). These deviations are most significant for sub-5 nm
particles. We herein report the reasons associated with the sensitivity of
survival probability to uncertainties, although there might be other causes
for these deviations that have been discussed in the literature (Kulmala
et al., 2017; Wang et al., 2020). Equation (3) shows that the survival probability
is a non-linear function of GR and CS, indicating that a small perturbation
in GR or CS will propagate into a large variation in the survival
probability. For example, the GR of sub-5 nm particles retrieved using the
mode-fitting method is systematically lower than the GR retrieved using the
appearance time method with an average ratio of 1:3. This difference is most
likely due to the underestimation of the mode-fitting GR (see “Methods” section), and
hence we compute the *P*_{theo} of sub-5 nm particles using the
appearance time GR. As shown in Fig. 6, this systematic difference in GR
with a factor of 3 corresponds to a difference in the median *P*_{theo} of >2 orders of magnitude.

To further quantify the sensitivity of the survival probability to
uncertainties, we plot the size-segregated survival probability as a
function of CoagS and GR. The size-dependent CoagS is characterized by the
condensation sink (CS) of sulfuric acid. As shown in Fig. 7, the survival
probability is most sensitive to uncertainties at small particle sizes and
high CS/GR values, under which conditions particles are mostly vulnerable to
coagulation scavenging. The sensitivity is herein defined as
−d log${P}_{\mathrm{theo}}/\mathrm{d}$(CS/GR), and it can be readily computed using Eq. (3). The
value of the sensitivity indicates the order of magnitude of the uncertainty
in *P*_{theo}. For instance, for a 1.4 nm particle with CS/GR =20 nm^{−1}, the sensitivity per nanometer growth is ∼1.6,
indicating that a ±10 % uncertainty in CS/GR will lead to an
uncertainty factor of $\mathrm{10}{}^{\mathrm{1.6}\times \mathrm{10}\phantom{\rule{0.125em}{0ex}}\mathit{\%}}=\mathrm{1.45}$ (equivalent to −31 %
or +45 % relative uncertainty) in the *P*_{theo} for particle growth from
1 to 2 nm. Similarly, the same ±10 % uncertainty in CS/GR will
lead to an uncertainty factor of 3.0 (equivalent to −67 % or +200 %
relative uncertainty) in the overall *P*_{theo} for particle growth from 1.4 to 100 nm. For urban Beijing, NPF is usually observed with high CS/GR
(Fig. 7a); hence the survival probability can be very sensitive to
uncertainties, especially for freshly nucleated particles in the sub-5 nm
size range under a high CS/GR value. For example, the sensitivity of
*P*_{theo} for particle growth from 1.4 to 100 nm is 12.0 for CS/GR =50 nm^{−1}, indicating that with a typical 100 % uncertainty in the
measured GR, the uncertainty in *P*_{theo} can be as high as 12 orders of
magnitude. Further discussions on the uncertainties in the CoagS and
survival probability can be found in Tuovinen et al. (2020, 2022).

In addition to the uncertainties in *P*_{theo}, there may be variations in
the measured survival probability due to the complex inhomogeneous
atmosphere. The homogeneity approximation as required by Eqs. (4)–(6) (see
“Methods” section) seems to be on average valid for regional NPF events in urban
Beijing, as indicated by the consistency of median measured and theoretical
survival probabilities. However, the growth of new atmospheric particles
from the cluster size to 100 nm takes hours (Fig. 6), during which period
the measured aerosol size distribution may be significantly affected by
transport. For some NPF events in urban Beijing, significant influences of
transport on the measured survival probability are sometimes observed, which
can be readily identified according to the abrupt changes in the measured
aerosol size distributions. Besides atmospheric inhomogeneity, traffic
emissions and other sources may add to the uncertainties in the measured
survival probabilities.

Figure A4 shows an NPF event measured at Hyytiälä as a case study
for the significant influence of NPF on the measured aerosol size
distributions. The measured mode d$N/$d log*d*_{p} increased with a growing
particle size until ∼11:00 LT (UTC+2), showing a high d$N/$d log*d*_{p}
region of new particles at ∼10 nm. Consequently, ${P}_{{n}_{\mathrm{log}}}$
and *P*_{n} for 7–12 nm particles were larger than 1.0. Particle accumulation
in a certain size range due to size-dependent particle growth rate was not
the main cause of the high d$N/$d log*d*_{p} region, as a clear pattern of rapid
particle growth can be seen from the growing mode. The wind direction was
relatively stable, though there was an increase in the wind speed at
∼10:00 LT (UTC+2). According to the analysis in Lampilahti et al. (2021), the high d$N/$d log*d*_{p} region and therefore the unphysical values of
${P}_{{n}_{\mathrm{log}}}$ and *P*_{n} were likely to be caused by vertical transport of new
particles. This vertical transport as an external source of particles is
supported by the increasing total concentration of particles in the growing
mode before 11:00 LT (UTC+2). Interestingly, *P*_{J} coincides with *P*_{theo}, though the
value *J* computed using 25 nm as an upper size limit did not necessarily
characterize the particle formation rate. For particles larger than 12 nm,
the trend of ${P}_{{n}_{\mathrm{log}}}$ followed that of *P*_{theo}, though there was still the
influence of transport on the measured aerosol size distributions. This case
study shows that for a certain NPF event, the measured survival probability
may be heavily influenced by the inhomogeneity of the atmosphere. Analyses
based on air homogeneity (e.g., backward trajectory), as well as statistical
analyses based on long-term measurements, may help us to reduce the
uncertainties in measured survival probabilities.

## 4.4 Implications on particle survival probability in measurements and models

The above analysis has shown that ${P}_{n{}_{\mathrm{log}}}$ can be used to approximate
the survival probability of particles in a growing population and that
*P*_{n} and *P*_{J} can be used for quasi-steady-state size distributions.
Compared to *P*_{J}, which is retrieved from particle formation rates,
${P}_{n{}_{\mathrm{log}}}$ and *P*_{n} can be readily obtained from the measured
aerosol size distributions. Further, the validity of *P*_{J} in atmospheric
NPF is usually limited to sub-10 nm particles, as the population balance
assumption for calculating the formation rate of large new particles (e.g.,
>25 nm) is challenged by transport and emissions. In contrast,
${P}_{n{}_{\mathrm{log}}}$ and *P*_{n} can provide the measured survival probabilities
of large particles (e.g., up to 100 nm). However, for the sub-5 nm size
range, *P*_{J} seems to be less sensitive to atmospheric variations than
${P}_{n{}_{\mathrm{log}}}$ and *P*_{n} (see Fig. 6).

The validity of ${P}_{n{}_{\mathrm{log}}}$, *P*_{n}, and *P*_{J} in different types of NPF
events calls for attention to retrieving the survival probability from
measurements and applying the survival probability in models. As we show in Fig. 2, applying *P*_{n} or *P*_{J} to a growing aerosol population
would significantly underestimate the survival probability. Alternatively,
improperly relating the formation rate and the survival probability might
significantly underestimate the formation rates of large particles and cloud
condensation nuclei. For example, Kerminen and Kulmala (2002) proposed
that the formation rate of critical clusters (e.g., 1 nm) can be derived
using the formation rate of the detected smallest particles (e.g., 3 nm) and
the theoretical survival probability. Assuming an accurateness of the CoagS
and GR for *P*_{theo} evaluation, the derived formation rate of critical
clusters is expected to be relatively accurate since the size distribution
of freshly nucleated particles is usually at a quasi-steady state. The
maximum systematic uncertainty in this derivation associated with the type
of NPF events is no more than 3 nm$/$1 nm =3. However, if one predicts
the formation rate of cloud condensation nuclei using *P*_{theo} and the
formation rate of critical clusters, the predicted formation rate may be
underestimated by 1–2 orders of magnitude as the size distributions of large
particles may not be at a steady state.

The sensitivity of *P*_{theo} to measurement uncertainties also calls for
accurate assessments of GR. We have shown that with carefully computed GR as
well as CoagS, the *P*_{theo} in urban Beijing is, on average, consistent
with the measured survival probability. However, if the GR was
underestimated by a factor of 3, the *P*_{theo} could be off by several
orders of magnitude (e.g., in Fig. 6), largely affecting the estimated
influences of NPF on the atmospheric environment. The consistency between
the measured median theoretical and measured survival probability also
provides a potential method to retrieve the median GR of new particles in
different atmospheric environments. Practically, it may be challenging to
accurately retrieve the GR from the measured aerosol size distributions or
gaseous precursors, while the median GR retrieved from the measured survival
probability may be used as a reference.

We reported methods to retrieve the survival probability of new atmospheric
particles from different types of new-particle-formation events and
investigated the consistency between the measured survival probability and
theoretical predictions. One new method based on the logarithmic size-scale
distribution function ${n}_{\mathrm{log}}=\mathrm{d}N/$d log*d*_{p} and two conventional methods
based on the new-particle-formation rate *J* and the linear size-scale
distribution function ${n}_{\mathrm{log}}=\mathrm{d}N/$d log*d*_{p} were tested. A sectional aerosol
kinetic model was used to generate simulated aerosol size distributions for
testing these methods. The theoretical survival probability against
coagulation scavenging predicted using the size-dependent coagulation sink
and growth rate was first validated using the definition of the survival
probability, and it was then used as a benchmark.

Based on the simulation results and theoretical analysis, we found that
*n*_{log} can be used to retrieve the measured survival probability from a
growing aerosol population with a relatively constant geometric standard
deviation. The influences of a size-dependent geometric standard deviation
can be readily accounted for using Eq. (7). *J* or *n* can be used for
quasi-steady-state size distributions that are significantly affected by the
continuous formation of new particles. Misapplying *J* or *n* to a growing aerosol
population will underestimate the survival probability. The above findings
were supported by measured new-particle-formation events in urban Beijing,
the Finnish boreal forest, and the CLOUD chamber. The size distribution of
sub-10 nm particles during NPF in urban Beijing was usually at a
quasi-steady state; hence the survival probability could be retrieved using
*n* and *J*. The validity of *n* and *J* for survival probability computation was also
found to be valid for the steady-state size distribution measured in a CLOUD
chamber experiment. For a test NPF event in the Finnish boreal forest and
the growth of particles larger than 25 nm for 65 NPF events in urban
Beijing, the survival probability could be retrieved using *n*_{log}.
Compared to the method based on *J*, the methods based on *n* and *n*_{log} have
advantages in their convenience in terms of computation, less sensitivity to
uncertainties, and the applicability to particles up to the cloud
condensation nucleus size (e.g., 100 nm).

We finally compared the measured survival probability in urban Beijing retrieved properly using the above methods and the theoretical survival probability against coagulation scavenging. For 65 NPF events obtained from long-term measurements, the measured and theoretical survival probabilities are on average consistent with each other in the 1.4–100 nm size range, though both are sensitive to measurement uncertainties and atmospheric variations.

Data are available via the link https://doi.org/10.5281/zenodo.6704909 (Cai, 2022). The code for simulating NPF is available in Li and Cai (2020).

RC designed the research; CD and JJ collected the Beijing data; DS, JK, and MK collected the Hyytiälä data; DS collected the CLOUD data; RC and CL prepared the model and performed the simulations; RC, JK, JG, VMK, and MK analyzed the data; RC wrote the paper with input from all co-authors.

The contact author has declared that none of the authors has any competing interests.

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

Technical and scientific staff in BUCT/AHL are acknowledged.

This research has been supported by the Academy of Finland (grant nos. 337549, 302958, 1325656, 311932, 316114, 332547, 325647, and 346370), the National Natural Science Foundation of China (grant nos. 22188102 and 92044301), the H2020 European Research Council (grant nos. 772206, 895875, and 764991), the Jane and Aatos Erkko Foundation (“Quantifying carbon sink, CarbonSink+ and their interaction with air quality”), Samsung PM_{2.5} SRP, and the Hungarian Research, Development and Innovation Office (grant no. K132254).

Open-access funding was provided by the HelsinkiUniversity Library.

This paper was edited by Markus Petters and reviewed by Vijay Kanawade and one anonymous referee.

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