The growth rate of atmospheric new particles is a key parameter that determines their survival probability of becoming cloud condensation nuclei and hence their impact on the climate. There have been several methods to estimate the new particle growth rate. However, due to the impact of coagulation and measurement uncertainties, it is still challenging to estimate the initial growth rate of new particles, especially in polluted environments with high background aerosol concentrations. In this study, we explore the influences of coagulation on the appearance time method to estimate the growth rate of sub-3 nm particles. The principle of the appearance time method and the impacts of coagulation on the retrieved growth rate are clarified via derivations. New formulae in both discrete and continuous spaces are proposed to correct for the impacts of coagulation. Aerosol dynamic models are used to test the new formulae. New particle formation in urban Beijing is used to illustrate the importance of considering the impacts of coagulation on the sub-3 nm particle growth rate and its calculation. We show that the conventional appearance time method needs to be corrected when the impacts of coagulation sink, coagulation source, and particle coagulation growth are non-negligible compared to the condensation growth. Under the simulation conditions with a constant concentration of non-volatile vapors, the corrected growth rate agrees with the theoretical growth rates. However, the uncorrected parameters, e.g., vapor evaporation and the variation in vapor concentration, may impact the growth rate obtained with the appearance time method. Under the simulation conditions with a varying vapor concentration, the average bias in the corrected 1.5–3 nm particle growth rate ranges from 6 %–44 %, and the maximum bias in the size-dependent growth rate is 150 %. During the test new particle formation event in urban Beijing, the corrected condensation growth rate of sub-3 nm particles was in accordance with the growth rate contributed by sulfuric acid condensation, whereas the conventional appearance time method overestimated the condensation growth rate of 1.5 nm particles by 80 %.

New particle formation (NPF) is frequently observed in various atmospheric
environments (Kulmala et al., 2004; Kerminen et al., 2018; Nieminen et
al., 2018; Lee et al., 2019). It contributes significantly to the number
concentrations of aerosols and cloud condensation nuclei (CCN) and hence
impacts the global climate (Kuang et al., 2009; Kerminen et al., 2012).
The new particle growth rate is one of the key parameters to characterize NPF
events. On the one hand, the newly formed particles (

Although new particle growth rates are frequently reported in various environments around the world, it remains difficult to retrieve accurate particle growth rates from an ambient dataset. Due to the varying atmospheric conditions, significant Kelvin effect, and size-dependent particle compositions, the particle growth rate is a function of both time and particle size. The measured evolution of aerosol size distribution does not directly indicate the size-and-time-resolved growth rate of single particles because one cannot directly track single particles from the size distributions. There are several methods to obtain the size-and-time-resolved growth rate by solving aerosol general dynamic equations (GDE; Kuang et al., 2012; Pichelstorfer et al., 2018). However, few applications of these GDE methods have been reported for particle growth analysis in the real atmosphere (e.g., Kuang et al., 2012). The most likely reason for this is that these GDE methods are sensitive to measurement uncertainties caused by atmospheric instability and instruments, which needs to be solved in future studies.

Apart from solving the GDEs, the widely used methods to estimate the particle growth rate are based on finding the representative particle diameter or time. The representative-diameter method usually uses the peak diameter of the size distribution of new particles and estimates its increase rate from its temporal evolution. The rate of increase in peak diameter is then taken as the particle growth rate (Kulmala et al., 2012) after correcting for (or sometimes neglecting) the influence of coagulation on the peak shifting (Stolzenburg et al., 2005). During the correction, coagulation is often classified into innermodal coagulation (self-coagulation) and intermodal coagulation (Anttila et al., 2010; Kerminen et al., 2018). The peak diameter is usually obtained by fitting a lognormal function to the measured aerosol size distribution of new particles. With a distinct peak diameter in the growing particle population, this method is theoretically feasible for estimating new particle formation rates. However, the mode fitting is usually tricky, especially when there is no well-defined mode in the growing distribution, due to either the aerosol distribution itself or the measurement uncertainties.

The representative-time method estimates the corresponding time for a series of diameters according to a certain criterion and then calculates the growth rate according to the relationship between the diameters and their corresponding time (Dada et al., 2020). The corresponding time is determined as the time to reach either the maximum concentration (maximum concentration method; Lehtinen and Kulmala, 2003) or a certain proportion of the maximum concentration (appearance time method; Lehtipalo et al., 2014) of a given particle size bin. Previous studies have tested the appearance time under various modeling conditions. Their results indicate that some appearance time methods can reproduce the theoretical growth rate within acceptable uncertainties under certain test conditions (Lehtipalo et al., 2014) but not under other test conditions (Olenius et al., 2014; Kontkanen et al., 2016; Li and McMurry, 2018). As shown in the “Theory” section below, the discrepancy is because the slope of particle size against appearance time usually convolves other information (e.g., coagulation) in addition to particle growth.

Determining the growth rate of sub-3 nm particles is more challenging than determining
that of larger particles. Firstly, there are considerable uncertainties in
the measured sub-3 nm aerosol size distributions (Kangasluoma
et al., 2020) compared to larger-sized particles (e.g.,

Coagulation impacts both particle growth and the growth rate calculation, especially for polluted environments and some chamber studies with high aerosol concentrations. The impact of coagulation on aerosol dynamics has been known for decades (e.g., McMurry, 1983). Recent studies have discussed the importance of considering coagulation when estimating the new particle growth rate (Cai and Jiang, 2017), the influence of transport on measured size distributions (Cai et al., 2018), and primary particle emissions (Kontkanen et al., 2020) under a high aerosol concentration. Similarly, neglecting particle coagulation may cause a bias in the retrieved particle growth rate. Therefore, the coagulation growth has to be considered before investigating the contributions of various condensing vapors to particle growth.

In this study, the feasibility and limitations of the appearance time method are investigated based on theoretical derivations. The impact of coagulation on the retrieved growth rate using the appearance time method is explored and then corrected for. Aerosol dynamic models are used to test the conventional and corrected methods. After that, the corrected appearance time method is applied in a typical NPF event in urban Beijing to show the impact of coagulation on growth rate evaluation in the real atmosphere. In addition, the potential uncertainties in the corrected appearance time method caused by vapor evaporation and the variation in vapor concentration are discussed.

Before deriving the formulae for the appearance time method, the definitions of particle growth and coagulation loss have to be clarified to avoid potential misunderstanding. Although widely used in NPF analyses, the exact meanings of these two concepts vary with their applied conditions.

The particle growth rate, by definition, is the rate of increase in particle
diameter as a function of time for a given particle. Assuming that there is
a sufficient number of particles of the same size and compositions, it is
reasonable to neglect the influence of the stochastic effect due to a low
particle number and use the expectation of the single-particle growth rate
to characterize the growth of the aerosol population with the same size.
When there is only one non-volatile condensing vapor, the formula for the
expectation of the single-particle condensation growth rate (referred to as
the condensation growth rate below for simplicity) is shown in Eq. (1):

In addition to the condensation of vapors, coagulation also contributes to
particle growth. For a given particle with the size of

When retrieving the particle growth rate from the measured aerosol size distributions, the retrieved value is named the apparent growth rate. “Apparent” emphasizes that the method does not necessarily guarantee that the retrieved growth rate is equal to the condensation or total growth rate of a single particle or the investigated aerosol population. When using the representative-diameter method, the retrieved apparent growth rate is the increasing rate of the peak diameter and does not directly characterize the growth of any particle(s). For instance, the coagulation loss rate is a function of particle diameter; as a result, the peak diameter shifts towards larger sizes with time because smaller particles are scavenged faster by coagulation than larger particles. Similarly, other size-dependent processes such as condensation and coagulation growth also cause the shift of peak diameter. As a result, the apparent growth rate sometimes needs to be corrected before it is taken as the total growth rate or the condensation growth rate (Stolzenburg et al., 2005). When using the representative-time method, although the retrieved apparent growth rate is close to the condensation growth rate under some modeling conditions (Lehtipalo et al., 2014), their deviation can be significant under other conditions, on which we elaborate in Sect. 4.

For a given particle, its coagulation with another particle can be
classified into coagulation growth and coagulation loss as aforementioned.
This classification is based on the Lagrangian specification that tracks the
growth of a single particle. In contrast, according to the Eulerian
specification that focuses on given particle diameters, each coagulation
causes a sink of two particles and a source of one new particle with a
larger diameter regardless of the particle sizes. Herein, we define the
coagulation sink and source as the loss and production rate for particle
size bins in the Eulerian specification. According to these definitions, the
coagulation of a large particle with another smaller particle is counted as
the coagulation sink (in the Eulerian specification) but not as the
coagulation loss (in the Lagrangian specification) of the large particle.
Following previous studies, we use CoagS (s

The impacts of coagulation characterized in the Lagrangian and Eulerian specifications.

The conventional appearance time method has been discussed in detail in
Lehtipalo et al. (2014), and its derivation under
ideal conditions has been reported in He et al. (2020). Here we briefly
describe the procedure to retrieve the particle growth rate from the temporal
evolution of a measured aerosol size distribution using the appearance time
method. For each aerosol size bin, its corresponding appearance time is
determined as the moment that the measured aerosol concentration in this
size bin reached 50 % or any other given proportion of its maximum
concentration during the event. The maximum and half-maximum concentration
of this size bin can be either taken from the smoothed temporal evolution of
the measured concentration or determined by fitting a sigmoid function to
the measured data. The growth rate is then estimated as the slope of the
diameter of aerosol size bins versus their corresponding appearance time;
i.e.,

The correction formulae for influences of coagulation on the appearance time
method in the discrete space are

When measuring aerosol size distribution using size spectrometers, the
measured distributions are usually reported in a certain number of sectional
bins. Therefore, in addition to the formula in the discrete form (Eq. 6),
the correction formulae for the appearance time in the sectional form are
given below:

The derivation for Eq. (6) is detailed in Sect. 4.1 and 4.2. The new and conventional appearance time methods are tested using a discrete-sectional model in Sect. 4.3 and a measured atmospheric NPF event in Sect. 4.4.

The proposed correction formulae for influences of coagulation on the appearance time method (Eqs. 6 and 7) are derived in Sect. 4.1 and 4.2. To test the validity of these corrections, numerical models are used to simulate an evolving aerosol size distribution and provide the theoretical growth rate according to the input monomer concentration. The growth rate is retrieved from the simulated aerosol size distribution using the conventional and corrected appearance time methods and then compared to the theoretical growth rate. Since the correction formulae are proposed only for the influence of coagulation, the influences of evaporation and varying vapor concentration on the retrieved growth rate with the appearance time method are discussed in Sect. 4.3. After that, the conventional and corrected appearance time methods are applied to a new particle formation event measured in urban Beijing. The impact of coagulation on the appearance time of new particles and hence growth rates are indicated by the differences between the growth rates retrieved from the measured aerosol size distributions using the conventional and corrected appearance time methods.

A discrete aerosol model and a discrete-sectional aerosol model based on
aerosol dynamics were used to provide an evolving aerosol size distribution
and hence to test the conventional and corrected appearance time methods.
The discrete model assumed that new particle formation is driven by the
nucleation and condensation of a certain single-component condensing vapor
(H

The principle of the appearance time method and the impact
of the external sink.

The discrete-sectional model was composed of 30 discrete bins (up to 2.2 nm) and 400 sectional bins (up to 230 nm). The vapor concentration was assumed to follow a normal distribution to simulate its diurnal variation in the real atmosphere. A growth enhancement factor as a function of particle size (Kuang et al., 2010) was used to account for the condensation of multiple vapors. A certain concentration of 100 nm particles was used as background particles, and their concentration and size were kept constant during each simulation. The discrete-section model was coded in MATLAB, and it is detailed in Li and Cai (2020).

The simulation conditions for Figs. 1 and 3–7 are summarized in Table 2. The simulations with varying vapor concentration are summarized in Table A1. Note that the aerosol dynamic models are only used to provide a benchmark to compare the conventional and corrected appearance time methods in this study.

The simulation conditions for Figs. 1 and 3–6. The symbol
“

The NPF event measured on 24 February 2018, in urban Beijing, was used to
test the influences of coagulation on the appearance time method. During
08:00–16:00 (local time, UTC

The measurement site is located on the west campus of the Beijing University of Chemical Technology, which is close to the west 3rd Ring Road of Beijing. The aerosol size distributions were measured using a homemade particle size distribution system (PSD; Liu et al., 2016) and a homemade diethylene glycol scanning mobility particle spectrometer (Jiang et al., 2011a; DEG-SMPS; Cai et al., 2017) equipped with a core-sampling apparatus (Fu et al., 2019). The sulfuric acid monomer and dimer concentrations were measured using a long chemical ionization time-of-flight mass spectrometer (ToF-CIMS; Aerodyne Research, Inc.; Jokinen et al., 2012). The meteorological data were measured using a local weather station (Vaisala, AWS310). More details on this measurement site and the instruments have been introduced elsewhere (Deng et al., 2020b).

Prior to investigating the impacts of coagulation on the appearance time
method, we briefly illustrate the principle of the appearance time method.
He et al. (2020) recently demonstrated that the appearance time method
can retrieve the condensation growth rate under ideal conditions. The ideal
conditions are as follows:

The vapor concentration is constant.

The initial concentrations of new particles are equal to zero.

Condensation is the only cause of the change in particle concentrations; i.e., there is no coagulation, evaporation, external loss, etc.

The condensation rate (i.e., coagulation rate between vapor and particles) is independent of the particle diameter.

Under the above ideal conditions, the population balance equation for a
particle containing

Solving the differential equations in Eq. (10) yields the analytical solution
for

Figure 1a shows the concentrations of

Equation (12) indicates that the slope of particle size in terms of its
molecule number versus its appearance time is approximately equal to its
condensation growth rate; i.e.,

The retrieved appearance time as a function of particle
size. The particle size is characterized by the number of molecules
contained in each single particle. The scatterplots are the appearance time
retrieved from the simulated concentrations. The curves are the approximate
solutions for the 50 % appearance time,

According to the derivations above, the 50 % size-resolved appearance time (referred to as 50 % appearance time for short) method can retrieve the particle growth rate under the given ideal conditions. The slope of particle size versus the appearance time is approximately equal to the condensation growth rate. That is, this slope is mainly determined by condensation growth under these ideal conditions. Note that Eq. (13) is only valid for the 50 % appearance time, whereas other thresholds to determine the appearance time may cause systematic bias. This bias comes from the non-parallelism of particle concentration curves (see Fig. 1). As shown in Fig. 2, in this test case, the 5 % appearance time method overestimates the growth rate by 15 % and the 95 % appearance time method underestimates the growth rate by 12 %. It should be clarified that since these biases are not huge, it is acceptable to use other thresholds instead of 50 % to reduce the impact of measurement uncertainties in practical applications.

Particle evaporation is assumed to be negligible in the above derivations,
yet Fig. 2 indicates that size-independent evaporation does not
significantly impact the validity of the 50 % appearance time method.
Assuming a size-independent evaporation rate,

Note that the equality between the slope of particle size versus the 50 % appearance time and the net condensation growth rate holds only under the above ideal conditions. The following derivations and results in Sect. 4.2 will show how the slope is affected by coagulation while maintaining the same condensation growth rate.

We first show the impact of an external sink on the appearance time method.
The external sink is herein referred to as the sink due to coagulation with
background particles, wall loss, dilution, transport, etc. For the
convenience of comparison with Fig. 1a, the external sink is assumed to be
temporally independent of the particle diameter. The impact of its size
dependency will be discussed later. Considering the constant external sink,
the population balance equation for

Similarly to Eqs. (11) and (12), the approximate solutions for

Equations (15) and (16) indicate that the impact of ES on

Combining Eqs. (1), (13), and (16), the impact of the external sink can be readily
corrected for. The correction formula is

The impact of sinks for the appearance time method and its
correction. The theoretical curve is obtained using the condensation rate of
the condensing vapor (Eq. 1) and the scatterplots are obtained using the
conventional and corrected appearance time method. The sink is assumed to be
independent of and dependent on particle diameter in

As shown in Fig. 3a, with a vapor concentration of

Practically, the coagulation coefficient (

As shown in Fig. 3b, when

The impact of coagulation sink (CoagS) due to colliding with a smaller particle on the appearance time method. Only the coagulation with a larger particle is accounted for in the correction using the background CoagS.

In addition to the coagulation with another background particle, the coagulation between two new particles also contributes to the CoagS of both these two particles. As explained in Sect. 2.2, no matter how small the coagulating particle is, the coagulation between a given particle and any other particle should be accounted for in CoagS. This is because the appearance time method is derived in the Eulerian specification and the CoagS is defined for a certain particle diameter rather than for a certain particle. In contrast, when focusing on the survival probability of new particles (Weber et al., 1997; Lehtinen et al., 2007), CoagS should be calculated in the Lagrangian specification; i.e., only the coagulation with a larger particle that causes particle loss should be accounted for. To emphasize the difference between the two definitions of CoagS, the corrected growth rates using the total (Eulerian) CoagS and the background (Lagrangian) CoagS are compared in Fig. 4. The particle source due to coagulation is not considered in this comparison. A constant concentration of 100 nm particles is used as the background particles. The background CoagS refers to the sink due to coagulation with all larger particles, including both the background particles and new particles. The total CoagS is calculated using Eq. (3). Note that due to the contribution of new particles, the total CoagS does not follow a simple decreasing trend with the increasing particle diameter (see Fig. B1 in Cai and Jiang, 2017). Hence, the empirical formula (Lehtinen et al., 2007) to generate a size-dependent CoagS in Fig. 4b should not be used for the total CoagS.

As shown in Fig. 4, the condensation growth rate after correcting for the
background CoagS is still overestimated. In contrast, the growth rate after
correcting for the total CoagS agrees with the theoretical growth rate for
particles larger than 1.3 nm. For sub-1.3 nm particles, the systematic bias
in the growth rate after correcting for the total CoagS is mainly caused by the
violation of the assumption that

In addition to CoagS, the impact of the coagulation source on the appearance
time also needs correcting for. Adding the coagulation source term to the
population balance equation of

Since CoagSrc

These three impacts of the coagulation source are accounted for in the corrected
appearance time method (Eqs. 6 and 7). Impact (1) is corrected for using Eq. (7).
To correct for impacts (2) and (3), we simply assume that CoagSrc

The impact of coagulation sink (CoagS) and coagulation source (CoagSrc) on the appearance time method and the contribution of coagulation growth. The terms cond and coag are short for condensation and coagulation, respectively. Note that both the solid and dashed lines are theoretical growth rates and their difference is the coagulation growth. Similarly, both the open and filled circles are the measured growth rates after correction and their difference is equal to the coagulation growth rate.

The corrected appearance time method was tested under various modeling
conditions. As the example in Fig. 5 indicates, the growth rates estimated
using the corrected appearance time method agree with the theoretical
growth rates. Equation (5) is able to retrieve the growth rate of sub-3 nm particles unless coagulation source is a governing reason for the
change in particle concentration; i.e., CoagSrc

In the above analysis, the correction for the influences of coagulation on the appearance time method was validated by derivations and simulations. However, there are still potential uncertainties in the corrected appearance time method because of other uncorrected influences. This section will discuss the uncertainties caused by vapor evaporation and a varying vapor concentration.

The impact of vapor evaporation on the appearance time
method.

Due to the uncorrected vapor evaporation, the appearance time method may
overestimate the growth rate for when the net condensation rate in the
Lagrangian specification (

As shown in Fig. 6b, the appearance time method corrected using Eq. (21) follows the theoretical net condensation growth rate. However, since Eq. (21) requires prior knowledge of the size-dependent evaporation rate, it is difficult to use this a correction for an atmospheric NPF event. Hence, one should note the uncertainties in the appearance time method for particles in the neighborhood of the critical size.

Note that the above discussions are for the case that nucleation and growth
are driven by a single volatile vapor. The condensation of a volatile vapor
onto a non-volatile particle may not introduce a significant bias to the
growth rate estimated using the appearance time method, as shown in Fig. S1
in the Supplement. This is because with an existing
non-volatile vapor,

The constant vapor concentration assumption may be valid for some chamber
experiments; however, the vapor concentration usually follows a diurnal
pattern in the real atmosphere. The varying vapor concentration may impact
the appearance time and hence the retrieved apparent growth rate. As
reported in previous studies (Lehtipalo et al., 2014; Olenius et al.,
2014), the retrieved appearance time is sensitive to the variation in vapor
concentration. In the presence of coagulation, it is difficult to correct for
the impact of the varying vapor concentration. Herein, we use the
discrete-sectional model to test the uncertainties in the corrected
appearance time method under a varying vapor concentration. The vapor
concentration is assumed to follow a normal distribution. The condensation
sink (10

The appearance time method under a varying vapor
concentration. The test condition is summarized in Table A1, no. 8.

In general, neglecting the variation in the vapor concentration introduces
biases to the appearance time method. As the example shown in Fig. 7 (test
no. 8 in Table A1) indicates, the deviation between the corrected particle
growth rate and the theoretical growth rate is smaller than the deviation
between the conventional growth rate and the theoretical growth rate.
However, for particles larger than

Although it is difficult to correct for the bias due to the varying vapor
concentration, one can try to avoid large uncertainties because the bias
seems to follow a certain pattern. Compared to the scenario of an increasing
vapor concentration, it is found that the discrepancy between the real and
retrieved growth rates are usually larger after the peak time of the vapor
concentration. As shown in Fig. 7, the appearance time of

For particles close to the size of vapor molecules (sub-2 nm in these tests), the appearance time usually convolves other information (e.g., the varying vapor concentration and the size-dependent coagulation coefficient) in addition to particle growth. Figure S2 shows that with larger vapor molecules, the size range for the discrepancy between the theoretical and retrieved growth rate shifts towards larger diameters. Considering the influences of vapor evaporation and varying vapor concentration on the appearance time method, one should be cautious about the size-resolved growth rate for particles close to the size of vapor molecules.

CoagS is assumed to be independent of time in the above discussions, whereas
it may vary significantly during an NPF event in the atmosphere. The varying
CoagS influences

In addition to vapor evaporation and the variation in vapor concentration, there may be other limitations for determining the appearance time in the atmosphere. Differently from controlled chamber studies (Dada et al., 2020), the uncertainties in atmospheric measurements pose challenges to growth rate estimation. These uncertainties come from instrumental biases; atmospheric turbulence; and the omitted contributions from transport, mixing, and emissions to the measured aerosol size distribution. Since the growth rate is calculated using a differential formula (Eq. 5), it is usually more sensitive to uncertainties than a physical quantity calculated using an integral formula (e.g., CoagS). For instance, the appearance time as a function of particle diameter in Fig. 7 had to be smoothed before calculating the growth rate using Eq. (5); otherwise, the calculated growth rate at some certain size bins would be negative. The applications of other methods to estimate the particle growth rate face the same challenge. As discussed in the Introduction, the appearance time method is used to estimate the new aerosol growth rate because other methods sometimes cannot report a growth rate in the concerned size range. Hence, further investigations concerning the uncertainties are needed for a better estimation of the growth rate in the atmosphere.

A case study for the appearance time method in the real
atmosphere.

A typical intense NPF event measured in urban Beijing is used to test the
correction for the influences of coagulation on the appearance time method.
During the event, the peak sulfuric acid concentration is

As shown in Fig. 8, the measured H

The impacts of particle coagulation are non-negligible compared to particle
growth and the growth rate calculation in urban Beijing. On one hand, the
conventional appearance time method overestimates the particle growth rate
for sub-3 nm particles in urban Beijing due to the impact of CoagS. The
deviation between the conventional and corrected growth rate decreases with
the increasing diameter because CoagS decreases with particle diameter. As
illustrated above, the correction for CoagSrc is only an approximation
rather than obtained based on solid derivations. However, the negligible
impact of CoagSrc on the measured growth rate in urban Beijing indicates
that this approximation does not cause significant bias. Differently from
coagulation growth which is weighted by particle size, the CoagSrc of

The difference between the measured and theoretical growth rates in Fig. 8
also indicates the growth mechanism of new particles. Considering the
uncertainties in the appearance time, the sum of condensation and
coagulation flux of sulfuric acid molecules and clusters is approximately
equal to the measured particle growth rate for

Summarizing all the analysis above, the growth rate retrieved using the conventional appearance time method may be systematically overestimated due to the impact of coagulation, especially for intensive NPF events in polluted environments. Such an overestimation may be significant for sub-3 nm particles because CoagS increases with decreasing particle size. In addition, the coagulation growth rate also needs to be corrected before investigating the condensation growth mechanism. For example, in the test case shown in Fig. 8, the retrieved condensation growth rate of 1.5 nm particles using the conventional appearance time method without correcting for the impact of CoagS and the coagulation growth rate is overestimated by 80 %. Figure 8 also indicates that the impact of CoagS may be negligible for larger particles and clean environments (see also Fig. S4). However, external sinks (e.g., dilution) may also cause an overestimation of the growth rate retrieved using the appearance time method if they are not properly corrected for.

The impact of coagulation on the particle growth rate retrieved using the appearance time method was investigated based on theoretical derivations and aerosol dynamics modeling. It was found that the often-used 50 % size-resolved appearance time method can reproduce the condensation growth rate only under the idealized condition without particle coagulation. When using the appearance time method in the real world, the coagulation sink, coagulation source, and coagulation growth need to be considered. Equations (5)–(9) provide a method in both discrete and sectional forms to correct for the impacts of the coagulation sink and coagulation source on the appearance time method. The feasibility of the correction for the influences of coagulation was verified using discrete and discrete-sectional aerosol models. In addition, vapor evaporation and the variation in vapor concentration was found to impact the appearance time method. The average uncertainties in the corrected 1.5–3 nm particle growth rate for each NPF event were 6 %–44 % in the test cases, and the maximum size-dependent uncertainty was 150 %. These uncertainties indicate that even after the correction for coagulation, one should be cautious about the appearance time method for particles close to the size of vapor molecules. Further, the growth rate of vapors and clusters is recommended to be estimated based on cluster dynamics instead of their representative time.

A typical NPF event measured in urban Beijing was used to show the quantitative impacts of coagulation on the retrieved growth rate. The systematic bias in the conventional appearance time method was observed for sub-3 nm particles due to the uncorrected impact of the coagulation sink. Besides, coagulation growth was non-negligible compared to the growth due to sulfuric acid condensation, which emphasizes the importance of distinguishing between the condensation and total growth rates. During the test event, the apparent growth rate of 1.5 nm particles retrieved using the conventional method was 80 % higher than the corrected condensation growth rate, whereas the corrected condensation growth rate was approximately equal to the theoretical growth rate contributed by sulfuric acid condensation.

Consider a particle population with a uniform diameter of

Comparing

The mean and maximum relative errors in the conventional
and corrected appearance time methods for 1.5–3 nm particles. The terms conv. and
corr. are short for the conventional and corrected methods, respectively.
The vapor concentration is assumed to follow a normal distribution with a
peak concentration of

The Taylor series of Eq. (A2) is

The Julia code for the discrete model is available upon request. The MATLAB code for the discrete-sectional model can be found via the link in Li and Cai (2020).

The measured data for Fig. 8 are available at

The supplement related to this article is available online at:

RC, JK, JJ, and MK initialized the study. RC and CL developed the models. RC and XCH derived the formulae. CD, YL, RY, CY, LW, and JJ performed the measurements and analyzed the data. RC performed the simulation and wrote the manuscript with the help of the other co-authors.

The authors declare that they have no conflict of interest.

This research has been supported by the Academy of Finland project (grant nos. 332547 and 1325656), a UHEL 3-year grant (grant no. 75284132), the National Key R&D Program of China (grant no. 2017YFC0209503), and the Samsung

This paper was edited by Radovan Krejci and reviewed by Wolfgang Junkermann, Nikolaos Kalivitis, and one anonymous referee.