Impacts of coagulation on the appearance time method for sub-3 nm particle growth rate evaluation and their corrections

The growth rate of atmospheric new particles is a key parameter that determines their survival probability to become 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 sub-3 nm particles, especially in polluted environments with high background aerosol concentrations. In this study, we explore the feasibility of the appearance time method to estimate the growth rate of sub-3 nm particles. The 20 principle of the appearance time method and the impacts of coagulation on the retrieved growth rate are clarified. New formulae in both discrete and continuous spaces are proposed to correct 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 to consider the impacts of coagulation on 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 non25 negligible compared to the condensation growth. Under the simulation conditions with a constant vapor concentration, the corrected growth rate agrees with the theoretical growth rates. The variation of vapor concentration is found to impact growth rate obtained with the appearance time method. Under the simulation conditions with a varying vapor concentration, the average bias of the corrected 1.5-3 nm particle growth rate range from 6-44%. 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 30 by sulfuric acid condensation, whereas the conventional appearance time method overestimated the condensation growth rate of 1.5 nm particles by 80%. https://doi.org/10.5194/acp-2020-398 Preprint. Discussion started: 15 May 2020 c © Author(s) 2020. CC BY 4.0 License.


Introduction
New particle formation (NPF) is frequently observed in various atmospheric environments (Kulmala et al., 2004;Nieminen et al., 2018;Lee et al., 2019). It contributes significantly to the number concentrations of aerosol and cloud condensation nuclei (CCN) and hence impacts the global climate (Kuang et al., 2009;Kerminen et al., 2012). New particle growth rate is one of the key parameters to characterize NPF events. On the one hand, the newly formed particles (~1 nm) 5 have to survive from coagulation scavenging before they grow to the CCN size (~100 nm). Given the same background aerosol concentration, i.e., the same coagulation loss rate, it is the growth rate that determines the survival probability of new particles (Weber et al., 1997;Lehtinen et al., 2007). Therefore, measuring new particle growth rate accurately contributes to understanding of the impact of NPF on the climate. On the other hand, particle growth rate is a key to investigate the growth mechanisms. Theoretical particle growth rates contributed by condensing vapors are usually compared to measured growth 10 rates to reveal the possible particle growth mechanisms (Ehn et al., 2014;Yao et al., 2018;Mohr et al., 2019). A non-biased and accurate determination of measured growth rates is an important fundament of these comparisons.
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, particle growth rate is a function of both time and particle size. The 15 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;. However, only few applications of these GDE methods have been reported for particle growth analysis in the real atmosphere (Kuang et al., 2012). The most likely reason is that these GDE methods are sensitive to measurement uncertainties 20 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 particle growth rate are based on finding the representing particle diameter or time. The representing diameter method usually uses the peak diameter of the size distribution of new particles and estimates its increase rate from its temporal evolution. The increase rate of peak diameter is then taken as particle growth rate  after correcting (or sometimes neglecting) the influence of coagulation on the peak 25 shifting (Stolzenburg et al., 2005). During the correction, coagulation is often classified into innermodal coagulation (selfcoagulation) 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 to estimate new particle formation rates. However, the mode fitting is usually tricky, especially when there is no well-defined mode in the growing distribution, either due to the aerosol 30 distribution itself or the measurement uncertainties.
The representing 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 https://doi.org/10.5194/acp-2020-398 Preprint. Discussion started: 15 May 2020 c Author(s) 2020. CC BY 4.0 License. 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 are able to reproduce the theoretical growth rate within acceptable uncertainties under certain test conditions  but not under other test conditions 5 (Olenius et al., 2014;Kontkanen et al., 2016;Li and McMurry, 2018). As shown in the Theory section below, the discrepancy is because that the slope of particle size against their 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 that of larger particles. Firstly, there are considerable uncertainties in the measured sub-3 nm aerosol size distributions  compared to larger-10 sized particles (e.g., > 10 nm, Wiedensohler et al., 2012). These uncertainties pose a great challenge to the methods based on solving aerosol general dynamic equations. Secondly, during a typical atmospheric NPF event, the sub-3 nm particle size distribution function usually decreases monotonically with the increasing diameter (Jiang et al., 2011b). As a result, the representing diameter method is usually difficult to cover the sub-3 nm size range. In contrast, despite lacking a clear mathematical understanding of the information convolved in the slope of appearance time against particle diameter, the 15 appearance time method is usually favored for sub-3 nm particles and clusters because of the existence of concentration peak of new particles during an atmospheric NPF event. In addition, the appearance time method is not significantly affected by the systematic instrumental uncertainties because the appearance time of each size bin is only determined by the relative signal rather than the absolute particle concentration.
Coagulation impacts both particle growth and the grow rate calculation, especially for polluted environments and some 20 chamber studies with high aerosol concentrations. The impact of coagulation on aerosol dynamics has been known since decades ago (e.g., McMurry, 1983). Recent studies discussed the importance of considering coagulation when estimating new particle growth rate , the influence of transport on measured size distributions (Cai et al., 2018), and primary particle emissions  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 25 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.
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 to growth rate evaluation in the 30 real atmosphere.

Particle growth rate
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. 5 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 condensing vapor, the formula for the expectation of the single-particle condensation growth rate (referred to as the condensation growth rate below for 10 simplicity) is shown in Eq. 1: where GRcond is the condensation growth rate (nm· s -1 ) that neglects evaporation, dp is particle diameter (nm), t is time (s), d1 is the diameter of the condensing vapor (nm), β1,p is the coagulation coefficient between d1 and dp (cm -3 · s -1 ), and N1 is the vapor concentration (cm -3 ). Particle evaporation is assumed to be negligible and the particle shape is assumed to be spherical both before and after the growth. Note that Eq. 1 is expressed in the discrete form, i.e., it does not assume a continuum particle 15 size (d1 → 0). When multiple vapors contribute to particle growth simultaneously, the total condensation growth rate is the sum of the condensation growth rates contributed by every single vapor.
In addition to the condensation of vapors, coagulation also contributes to particle growth. For a given particle with the size of dp, the coagulation with a particle much smaller than dp is usually considered as a contribution to its growth. In contrast, the coagulation with a particle much larger than dp is usually considered as the coagulation loss of particle dp. We follow this 20 convention to distinguish coagulation growth and loss, i.e., particle coagulation with another particle no larger than itself is taken as coagulation growth and otherwise, it is taken as coagulation loss. Hence, the formula for the expectation of singleparticle coagulation growth rate (referred to as the coagulation growth rate for simplicity) in the discrete form is: where GRcoag is the coagulation growth rate (nm· s -1 ), dmin is the minimum particle size (nm), βp,i is the coagulation coefficient (cm -3 · s -1 ) between dp and di, and Ni is the concentration (cm -3 ) of particles with the size di. Since both condensation and 25 coagulation contribute to particle growth, the total single-particle growth rate is equal to the sum of GRcond and GRcoag.
When retrieving particle growth rate from the measured aerosol size distributions, the retrieved growth rate is 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 representing https://doi.org/10.5194/acp-2020-398 Preprint. Discussion started: 15 May 2020 c Author(s) 2020. CC BY 4.0 License. diameter method, the retrieved apparent growth rate is the increase rate of the peak diameter and it 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 taken as the total growth rate or the condensation 5 growth rate (Stolzenburg et al., 2005). When using the representing time method, although the retrieved apparent growth rate is close to the condensation growth rate under some modeling conditions , their deviation can be significant under other conditions, on which we elaborate in section 4.

Coagulation sink and source
For a given particle, its coagulation with another particle can be classified into coagulation growth and coagulation loss as 10 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 particle with another smaller particle is counted as the coagulation sink (in the Eulerian 15 specification) but not as the coagulation loss (in the Lagrangian specification). In accordance with previous studies, we use CoagS (s -1 ) to represent the sink coefficient (Kulmala et al., 2001) and CoagSrc (cm -3 · s -1 ) to represent the production rate due to coagulation (Kuang et al., 2012). Their formulae in the discrete form are given below: where dmax is the maximum particle diameter (nm); dj is defined by dj 3 = dp 3 − di 3 ; Ni and Nj are the concentrations of di and dj, respectively; and other variables have been introduced above. Note that CoagS and CoagSrc are defined differently and their 20 units are also different. Table 1 summarizes the differences between the definitions in the Lagrangian and Eulerian specifications. Table 1 2

.3 Formulae for the new appearance time method
The correction formulae for the appearance time method in the discrete space is: 25 https://doi.org/10.5194/acp-2020-398 Preprint. Discussion started: 15 May 2020 c Author(s) 2020. CC BY 4.0 License.
GR corr,cond = GR corr,tot − GR coag (7) where GRcorr,tot is the total growth rate (nm· s -1 ) after correcting the impact of both coagulation sink and source; GRconv is the total growth rate (nm· s -1 ) retrieved by the conventional appearance time method; GRcorr,cond is the condensation growth rate (nm· s -1 ) after correction; GRcoag is the coagulation growth rate (nm· s -1 ); CoagS (s -1 ) and CoagSrc (cm -3 · s -1 ) are the coagulation sink and coagulation source term for dp, respectively; Np is the number concentration of particles with the size dp at its appearance time; Δdp (nm) is the size difference between two adjacent measured size bins, and Δt (s) is the time difference of 5 the appearance time of these two size bins. Note that coagulation growth is corrected in Eq. 7 but not Eq. 5.
When measuring particle 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. 5), the correction formula for the appearance time in the sectional form is given below: where dp,u (nm) and dp,l (nm) are the upper and lower size limits of a given size bin; dp (nm) is the representative diameter 10 (usually the geometric mean diameter) of this size bin; CoagS is the coagulation sink (s -1 ) for dp; CoagSrc is the coagulation source term (cm -3 · s -1 ) for the given size bin; N[dp,1 dp,u] is the measured concentration (cm -3 ) of the size bin [dp,l, dp,u] at tp; d1 is the diameter (nm) of the condensing vapor; and ni is the aerosol size distribution function (dni/dlogdi, cm -3 ) for the given size di.
The derivation for Eq. 5 is detailed in sections 4.1 and 4.2. The new and conventional appearance time methods are tested 15 using a discrete-sectional model in section 4.3 and a measured atmospheric NPF event in section 4.4.

Numerical models
A discrete aerosol model and a discrete-sectional aerosol model were used 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 20 certain single-component condensing vapor. The vapor concentration was set as a constant. Condensation, coagulation, and external loss were considered in this discrete model. The concentrations of particles up to the size of 100 vapor molecules https://doi.org/10.5194/acp-2020-398 Preprint. Discussion started: 15 May 2020 c Author(s) 2020. CC BY 4.0 License.
(~3.5 nm) were numerically solved using Julia. The theoretical condensation and coagulation growth rates were calculated using Eqs. 1 and 2.
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 a particle size (Kuang et al., 2010) was used to account for the condensation of 5 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 3-6 are summarized in Table 2. The simulations with varying vapor concentration are summarized in Table A1. 10 Table 2 3

.2 Measurements
The NPF event measured on Feb. 24 th , 2018, in urban Beijing was used to test the appearance time method. The measurement site locates 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 15 (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 spectrometers (ToF-CIMS, Aerodyne Research Inc., Jokinen et al., 2012). More details on this measurement site and the instruments have been introduced elsewhere . 20

Appearance time method under ideal conditions
Prior to investigating the impacts of coagulation on the appearance time method, we briefly illustrate the principle of the appearance time method. It can be demonstrated that the appearance time method is able to retrieve the condensation growth rate under ideal conditions. The ideal conditions are: 25


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 particle diameter. 30 Under such ideal conditions, the population balance equation for a particle containing i molecules is: https://doi.org/10.5194/acp-2020-398 Preprint. Discussion started: 15 May 2020 c Author(s) 2020. CC BY 4.0 License.
where N1, Ni-1, and Ni are the concentrations (cm -3 ) of the condensing vapor and particles containing i and i-1 monomer molecules, respectively; t is time (s); β is the coagulation coefficient (cm -3 · s -1 ) between a vapor molecule and any particle, which is assumed to be independent of the particle size in Eq. 10. For the case i = 2, the last term in Eq. 10 should be modified as 0.5βN1 2 .
Solving the differential equations in Eq. 10 yields the analytical solution for Ni: 5 where Ni,∞ is the concentration limit (cm -3 ) of Ni when t approaches infinite (dNi/dt = 0) and it is equal to 0.5βN1 2 (for i > 1) under these ideal conditions. Figure 1a shows the concentrations of Ni normalized by dividing by their corresponding Ni,∞. It can be seen that the distance between two adjacent concentration curves is approximately a constant though these curves are not parallel. Hence, the appearance time method takes the moment that Ni reaches a certain percentage of its maximum value (Ni,∞) as its 10 representative time. Previous studies indicate that the 50% size-resolved appearance time method which chooses the certain percent as 50% is more robust against non-ideal conditions compared to using the criterion of other percent values . As shown in Fig. 2, an approximate solution of t for Ni(t) = 0.5Ni,∞ (referred to as ti) is: 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., 15 where GRcond,n is the condensation growth rate in terms of the molecule number (s -1 ). The relationship between GRcond,n herein and GRcond in Eq.1 (which is defined with respect to particle diameter) is GRcond = GRcond,n×Δdi, where Δdi (nm) is the increase of di due to the condensation of one vapor molecule.

Figure 1
According to the derivations above, the 50% size-resolved appearance time (referred to as 50% appearance time for short) 20 method is able to retrieve 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 25 95% appearance time method underestimates the growth rate by 12%. It should be clarified that since these biases are not huge, https://doi.org/10.5194/acp-2020-398 Preprint. Discussion started: 15 May 2020 c Author(s) 2020. CC BY 4.0 License.
it is acceptable to use other thresholds in addition to 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 evaporation does not significantly impact the validity of the 50% appearance time method. Assuming a size-independent evaporation rate, E (s -1 ), ti is approximately equal to (ln2+i)/(βN1-E), which is bigger than that without evaporation. Meanwhile, considering evaporation, 5 the net condensation growth rate is equal to vapor condensation rate subtracted by particle evaporation rate, i.e., βN1-E. That is, the increase in appearance time agrees with the decrease in the net condensation growth rate. In practice, particle evaporation rate is usually size dependent due to the significant Kelvin effect. The bias caused by this size dependency of evaporation is similar to that of coagulation, which will be shown to have a minor effect below. In addition, the evaporation flux (cm -3 s -1 ) of a given particle containing i molecules is determined by Ni+1 rather than Ni. As a result, the slope of particle size versus the 10 50% appearance time may deviate from the net condensation growth rate when i is a small value and there is a non-negligible difference between Ni+1 and Ni.

Figure 2
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 will show how the slope is affected by 15 coagulation while maintaining the same condensation growth rate.

The impacts of coagulation and their corrections
We first show the impact of an external sink to 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 particle diameter. The impact of its size dependency will be 20 discussed later. Considering the constant external sink, the population balance equation for Ni is: where ES is the external sink (s -1 ) and other variables have been introduced in Eq. 10.
Similarly to Eqs. 11 and 12, the approximate solutions for Ni and its corresponding appearance time (ti) are: Equations 15 and 16 indicate that the impact of ES to ti is mathematically equivalent to vapor condensation (βN1). In the presence of a non-negligible external sink, the particle concentration will approach its limit faster than the scenario without 25 external sink (Fig. 1b). As a result, the slope of particle diameter versus appearance time is affected by both condensation growth and external sink. Combining Eqs. 1, 13, and 16, the impact of external sink can be readily corrected. The correction formula is: where GREScorr is the growth rate (nm· s -1 ) after correcting the external sink, GRconv is the growth rate (nm· s -1 ) retrieved by the conventional appearance time method, and d1 is the diameter (nm) of the condensing vapor.
As shown in Fig. 3a, with a vapor concentration of 5×10 6 cm -3 and an external sink ranging from 1×10 -3 to 5×10 -3 s -1 , the conventional appearance time method overestimates the condensation growth rate substantially. Such an overestimation caused 5 by mistaking external sink for condensation growth was also reported in previous studies (Olenius et al., 2014;Li and McMurry, 2018). In contrast, the corrected growth rate agrees well with the theoretical condensation growth rate.
Practically, the coagulation coefficient (β) and ES are functions of the particle diameter. For the convenience of illustration, we use the size-dependent coagulation sink, CoagS, as an example to represent the total particle sink due to coagulation, wall loss, dilution, and transport. Similarly to Eq. 15, the approximate analytical solution for Ni with the size-dependent β and 10 CoagS is: where β1,i (or β1,g) is the coagulation coefficient between a vapor molecule and a particle containing i (or g) molecules (cm -3 · s -1 ); CoagSi (or CoagSg) is the coagulation sink of particles containing i (or g) molecules (s -1 ); and other variables have been introduced above. Correspondingly, the ES term in Eq. 17 should be replaced with CoagSi to correct the impact of the sizedependent coagulation sink. When deriving Eq. 18, it is assumed that β1,iN1+CoagSi is close to β1,i-1N1+CoagSi-1. This 15 approximation is reasonable because both β1,i and CoagSi change gradually with the particle size, yet it introduces minor systematic biases in Ni and its corresponding appearance time.
As shown in Fig. 3b, when β1,i and CoagSi are size dependent, the corrected appearance time method is still able to reproduce the condensation growth rate. It is assumed that the CoagSi in Fig. 3b is contributed by only the large background particles. Hence, the CoagSi in Fig. 3b is estimated from the condensation sink (CS) using an empirical formula (Eq. 8 in 20 Lehtinen et al., 2007), where CS indicates the condensation loss rate of the vapor.

Figure 3
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 section 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 25 appearance time method is derived in the Eulerian specification and the CoagS is defined with respect to a certain particle diameter rather than with respect to 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 https://doi.org/10.5194/acp-2020-398 Preprint. Discussion started: 15 May 2020 c Author(s) 2020. CC BY 4.0 License.
of CoagS, the corrected growth rates using the total (Eulerian) CoagS and the background (Lagrangian) CoagS are compared in Fig. 4. Particle source due to coagulation is not considered in this comparison. A constant concentration of 100 nm particles 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  5 in . 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 the background CoagS is still overestimated. In contrast, the growth rate after correcting 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 of the growth rate after correcting the total CoagS is mainly caused by the violation 10 of the assumption that β1,iN1+CoagSi is close to β1,i-1N1+CoagSi-1. The size dependence of particle coagulation coefficient under the influence of new particle coagulation increases with decreasing particle size. For instance, under the test conditions, (β1,4N1+CoagS4)/(β1,3N1+CoagS3) = 1.12 while (β1,50N1+CoagS50)/(β1,49N1+CoagS49) = 1.01. As a result, the corrected appearance time method still overestimates the growth rate for sub-1.3 nm particles.

Figure 4 15
In addition to CoagS, the impact of coagulation source on the appearance time also needs correction. Adding the coagulation source term to the population balance equation of Ni yields: where CoagSi is the coagulation sink (s -1 ) corresponding to Ni, CoagSrci is the coagulation source term (cm -3 · s -1 ) corresponding to Ni, and other variables have been introduced above.
Since CoagSrci is determined by the concentrations of all particles containing 2 to i-2 molecules, it is difficult to obtain 20 an (approximate) analytical solution of Eq. 19. Here we provide an approximation method to correct the impact of coagulation source to the appearance time. Compared to the scenario that coagulation source is neglected, the coagulation source has three impacts on particle growth and its calculation: 1) the coagulation with a smaller particle contributes to particle growth; 2) the coagulation source increases the maximum particle concentrations; 3) the coagulation source shortens the time for particles to reach their maximum concentration. 25 Impact 1) can be readily corrected using Eq. 7. To correct impacts 2) and 3), we simply assume that CoagSrci is a constant during the increasing period of Ni and use the following formulae (see the Appendix for its derivation) to estimate the growth rate. The correction formula has been given in Eq. 5.
The corrected appearance time method (Eqs. 5-7) was tested under various modeling conditions. As the example in Fig.   5 indicates, the growth rates estimated using the corrected appearance time method agrees with the theoretical growth rates. 30 https://doi.org /10.5194/acp-2020-398 Preprint. Discussion started: 15 May 2020 c Author(s) 2020. CC BY 4.0 License.

Equation 5
is able to retrieve the growth rate of sub-3 nm particles unless when coagulation source is a governing reason for the change of particle concentration, i.e., CoagSrc/2Np is comparable or larger than β1,pN1. Under these conditions, CoagSrc may not be a constant and, hence, the approximation of CoagSrc/2Np may cause a bias. Fortunately, CoagSrc usually decreases with the increasing particle size due to the decreasing particle concentration. Furthermore, it will be shown in section 4.4 that CoagSrc does not have a major impact on the apparent growth rate even during an intensive atmospheric NPF event. Hence, 5 we consider Eq. 5 as a rough but sufficient formula to correct the impact of coagulation on the appearance time in the real atmosphere.

The impact of varying vapor concentration
In the above analysis, the vapor concentration is assumed to be constant over the whole particle growth period. This assumption 10 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 , the retrieved appearance time is sensitive to the variation of vapor concentration. In the presence of coagulation sink and coagulation growth, it is difficult to correct the impact of the varying vapor concentration. Herein, we use the discrete-sectional model to test the uncertainties of the corrected appearance time 15 method under a varying vapor concentration. The vapor concentration is assumed to follow a normal distribution. The condensation sink is contributed simultaneously by a certain number of 100 nm background particles and the new particles.
The growth rate is firstly estimated using the 50% size-resolved appearance time method and then corrected using Eq. 8. Since the vapor concentration varies with time, the retrieved growth rate characterizes particle growth at both different diameters and different time instead of the size-dependent growth at a certain moment. To keep in accordance with the appearance time 20 method, the theoretical condensation and coagulation growth rates of each dp are calculated at its corresponding tp.
In general, neglecting the variation of the vapor concentration introduces biases to the appearance time method. As the example shown in Fig. 6 (test No. 8 in Table) indicates, the corrected particle growth rate agrees with the theoretical growth rate better than the conventional growth rate. However, for particles larger than ~5 nm and smaller than ~2 nm, the appearance time method overestimates particle growth rate even after correcting the impact of coagulation in the test case. As summarized 25 in Table A1, the relative discrepancy depends on the exact conditions. The average discrepancy of the corrected appearance time method for 1.5-3 nm particles ranges from 6% to 44% in the test conditions, which is smaller than that of the conventional method.
Although it is difficult to correct 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 30 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. 6, the appearance time of ~4.9 nm particles is 12 h and a substantial discrepancy between the theoretical and retrieved growth rate is observed for particles larger than 4.9 nm. Fortunately, during a typical atmospheric NPF event, new particles usually grow large before the vapor concentration starts to decrease. To reduce this systematic bias, we suggest using the other methods, e.g., the representing diameter method to estimate particle growth rate when the vapor concentration 35 decreases. For sub-2 nm particles, the appearance time usually convolves other information (e.g., the varying vapor https://doi.org /10.5194/acp-2020-398 Preprint. Discussion started: 15 May 2020 c Author(s) 2020. CC BY 4.0 License.
concentration and the size-dependent coagulation coefficient) in addition to particle growth. Hence, one should be cautious about the sub-2 nm size-resolved growth rate.

Figure 6 4.4 Application in atmospheric measurements
A typical intense NPF event measured in urban Beijing is used to test the conventional and corrected appearance time 5 methods. During the event, the peak sulfuric acid concentration was ~6×10 6 cm -3 and the average CS for sulfuric acid was 0.024 s -1 . The theoretical condensation and coagulation growth rates are calculated using the measured sulfuric acid concentration and aerosol size distribution, respectively. Particle growth due to the uptake of sulfuric acid dimers is herein accounted as the condensation growth. Note that the sum of theoretical condensation and coagulation growth rate is not necessarily equal to the theoretical total growth rate for the measured NPF event. This is because only the condensation of 10 sulfuric acid is considered whereas other vapors may also contribute to new particle growth. The enhancement due to Van der Waals force is considered when calculating the coagulation coefficient (Alam, 1987;Chan and Mozurkewich, 2001;Stolzenburg et al., 2019). The appearance time retrieved from the measured aerosol size distributions was smoothed before estimating the particle growth rate.
As shown in Fig. 7, the impacts of particle coagulation are non-negligible compared to particle growth and the grow rate 15 calculation in urban Beijing. On one hand, the conventional appearance time method overestimates 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 a significant bias. 20 Different from coagulation growth which is weighted by particle size, the CoagSrc of dp is only determined by the number concentrations of particles smaller than dp (and their coagulation coefficient). Even under such an intense NPF event (with the maximum formation rate exceeding 200 cm -3 · s -1 ), the new particle concentration is usually much smaller than the vapor concentration due to the high CoagS and possibly cluster evaporation. Hence, it is sometimes acceptable to neglect the CoagSrc/Np term in Eqs. 5 and 8 to facilitate calculation. On the other hand, the coagulation with smaller particles enhances 25 particle growth and this enhancement increases with the increasing particle size. This emphasizes that during an intensive NPF event with a high new particle concentration, the condensation growth rate contributed by condensing vapors cannot be taken as the total growth rate that determines the survival probability of new particles.

Figure 7
The difference between the measured and theoretical growth rates in Fig. 7 also indicates the growth mechanism of new 30 particles. Considering the uncertainties of 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 ~3 nm particles, which indicates that sulfuric acid is a governing species that contribute to the initial growth of sub-3 nm particles during the test event. The deviation between the measured growth and theoretical growth for particles larger than ~3 nm indicates that there are other chemical species in addition to sulfuric acid (and the bases to neutralize it) contributing to particle growth. Note that the above discussion 35 is only based on a single case study. Hence, further investigations based on long-term measurements are needed to reveal the growth mechanism in the polluted atmospheric environment.
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 the 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. 7, the retrieved condensation growth rate of 1.5 nm particles using the conventional 5 appearance time method without correcting the impact of CoagS and the coagulation growth rate is overestimated by 80%. Figure 7 also indicates that the impact of CoagS may be negligible for larger particles and clean environments (see also Fig.  A1). 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.

Conclusions 10
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 is able to reproduce the condensation growth rate only under the idealized condition without particle coagulation.
When using the appearance time method in the real world, 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 the impacts of coagulation 15 sink and coagulation source to the appearance time method. The feasibility of the corrected method was verified using discrete and discrete-sectional aerosol models. In addition, the variation of vapor concentration was found to impact the appearance time method. The average uncertainties of the corrected 1.5-3 nm particle growth rate for each NPF event were 6-44% in the test cases, respectively. 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 of the conventional appearance time method was observed for 20 sub-3 nm particles due to the uncorrected impact of the coagulation sink. In addition, coagulation growth was non-negligible compared to the growth due to sulfuric acid condensation, which emphasizes the importance to distinguish 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. 25 Figure A1 Derivation of Eq. 1 30 Consider a particle population with a uniform diameter of dp (nm) and a concentration of N0. N0 is assumed to be sufficiently large so that the stochastics in particle growth is negligible. At the initial moment t0 (s), the mean particle diameter is dp. During a short time interval dt (s), β1,pN1N0dt particles collide with the condensing vapor with a diameter of d1, where β1,p is the https://doi.org /10.5194/acp-2020-398 Preprint. Discussion started: 15 May 2020 c Author(s) 2020. CC BY 4.0 License. coagulation coefficient (cm 3 · s -1 ) and N1 is the vapor concentration. Hence, the mean diameter ( p ̅̅̅ ) weighted by particle number concentration at the moment t0+dt is:

Table A1
Comparing p ̅̅̅ ( 0 ) and p ̅̅̅ ( 0 + d ) yields the condensation growth rate: The Taylor series of Eq. A2 is: where [ 1,p 1 ( 1 p ⁄ ) 6 ] is the Peano form of the remainder. The first term on the right-hand side of Eq. A3 is the formula 5 for particle growth rate in the continuous form and the second term (the remainder) is the difference between the grow rate formula in the continuous and discrete forms (Olenius et al., 2018). When dp is sufficiently larger than d1, Eq. A2 is reduced to 1,p 1 1 3 (3 p 2 ) ⁄ .

The impacts of coagulation source and their corrections 10
In this section, we present an approximate derivation for Eq. 5. For the convenience of illustration, particle size and growth rate are characterized using the molecule number rather than particle diameter. Assuming that condensation is the only cause of the change in Ni (Eq. 10), the apparent growth rate is equal to the condensation growth rate, i.e., GR conv = GR app (10) = 1,i 1 (Eq. A4) where GRapp (10) is the apparent growth rate (s -1 ) of particles containing i molecules and the superscript (10) indicates the population balance assumption in Eq. 10; β1,i is the coagulation coefficient (cm -3 · s -1 ) between a vapor molecule and particle i; 15 and Ni is the concentration (cm -3 ) of particle i. N1 is assumed to be a constant. The conventional appearance time method takes GRapp as the growth rate (GRconv) without correction. The source and maximum concentration of Ni are given below: where Src is source for Ni; the Ni-1,∞ is the maximum concentration of Ni-1 (at t → +∞); Ni-1 is the concentration of particle i-1 at its appearance time, hence, it is equal to 50% of the maximum concentration of Ni-1. where CoagSi is the coagulation sink (s -1 ) corresponding to Ni; and CoagSrci is the coagulation source term (cm -3 · s -1 ) corresponding to Ni.
As illustrated in the main text, CoagSi and CoagSrci change both Src and Ni,∞. The conventional (apparent) growth rate under this scenario can be obtained by accounting for these two aspects, i.e., i,∞ Code availability. The Julia code for the discrete model is available upon request. The Matlab code for the discrete-sectional model is publicly available in Li and Cai (2020).
Competing interests. The authors declare that they have no conflict of interest. Coagulation source (CoagSrc) for the next bin Table 2 The simulation conditions for Figs. 1 and 3-6. The symbol "√" indicates "yes" and the blank indicates "no".

Figure
No.

Figure 1
The principle of the appearance time method and the impact of the external sink. (a) Normalized particle concentrations as a function of time. The concentrations are normalized by dividing their corresponding maximum concentrations. The number concentration of the condensation vapor is assumed to be constantly 5×10 6 cm -3 . Particle coagulation sink and other sinks are assumed to be negligible. Particle size is indicated by the molecule number contained in 5 every single particle. The open scatters indicate the 50% appearance time corresponding to each particle size. (b) A constant external sink of 1.5×10 -3 s -1 is considered and other simulation conditions are the same as a). Note that due to the assumption of size-independent coagulation coefficient, the appearance time in this figures deviates from that in real new particle formation events.
10 Figure 2 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 scatters are the appearance time retrieved from the simulated concentrations. The curves are the approximate solutions for the 50% appearance time, ti = k×(i+ln2), where k is the slope of the curve (see Eqs. 7 and 11). β is the coagulation coefficient (cm 3 · s -1 ) between vapor and particles, N1 is the vapor concentration (5×10 6 cm -15 https://doi.org/10.5194/acp-2020-398 Preprint. Discussion started: 15 May 2020 c Author(s) 2020. CC BY 4.0 License. Figure 3 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 scatters are obtained using the conventional and corrected appearance time method. The sink is assumed to be independent and dependent of particle diameter in (a) and (b), respectively. The scatters for the corrected method lie on top of each other. The size-dependent coagulation sink in (b) was estimated from the condensation sink (CS) shown in legend using an empirical formula (Lehtinen et al., 2007). The coagulation sink is taken as 10 the input value of the model, hence, the validity of the empirical formula does not affect the accuracy of simulated size distribution or the growth rate. The minor discrepancy among the corrected growth rate comes from the size-dependent particle coagulation coefficient and coagulation sink.

Figure 4
The impact of coagulation sink (CoagS) due to colliding with a smaller particle to the appearance time method. Only the coagulation with a larger particle is accounted for in the correction using the background CoagS.

Figure 5
The impact of coagulation sink (CoagS) and coagulation source (CoagSrc) to the appearance time method and the 5 contribution of coagulation growth. 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.
10 Figure 6 The appearance time method under a varying vapor concentration. The test condition is summarized in Table A1, No. 8. (a) An NPF event simulated using a discrete-sectional aerosol dynamic model. The vapor concentration is assumed to follow a normal distribution (with background value of 10 5 cm -3 ). The 100-nm background particles are not shown. The particle diameter as a function of the appearance time is shown in the solid line. (b) The theoretical and retrieved particle growth rates. Cond and coag are short for condensation and coagulation, respectively. 15 https://doi.org/10.5194/acp-2020-398 Preprint. Discussion started: 15 May 2020 c Author(s) 2020. CC BY 4.0 License.

Figure 7
A case study for the appearance time method in the real atmosphere. (a) Aerosol size distribution and H2SO4 concentration during the test NPF day. The event was measured on Feb. 24 th , 2018, in urban Beijing. (b) Measured growth rates using the conventional and corrected appearance time methods and the theoretical growth rate contributed by sulfuric acid condensation and particle coagulation. Cond and coag are short for condensation and coagulation, respectively. Note that 5 the theoretical growth rate considers only the sulfuric acid condensation, hence, it may underestimate the overall condensation growth rate contributed by multiple condensing vapors.

Figure A1
Error of the growth rate retrieved using the conventional appearance time method for (a) 1.5 nm and (b) 5 nm 10 particles. The relative error is defined as (GRconv-GRcond)/GRcond, where GRconv is the growth rate retrieved by the conventional appearance time method and GRcond is the condensation growth rate. Coagulation growth is neglected in this figure. The approximate range of condensation sink in Beijing (Wang et al., 2013) and Hyytiä lä (Dal Maso et al., 2002) are marked with arrows, which indicate the typical condensation sink in polluted and clean environments, respectively. https://doi.org/10.5194/acp-2020-398 Preprint. Discussion started: 15 May 2020 c Author(s) 2020. CC BY 4.0 License.