Limited by computational quantum chemistry calculation results, SA–DMA nucleation is commonly simulated in the range of clusters containing not more than four SA or four DMA molecules (Olenius et al., 2013; Ortega et al., 2012). As unstable clusters would evaporate with higher rates, the formation of larger clusters potentially follows the pathways of the most stable clusters. In addition, as the SA–DMA clusters are increasingly stable along the main pathway of cluster formation, the clusters not smaller than A4B4 (hereafter AmBn refers to clusters containing m SA and n DMA molecules) are assumed to not evaporate back in these simulations. Although there are uncertainties in the pathways presented, based on different quantum chemistry methods, it is well accepted that the AmBm (m=1 to 4) and A2B1 clusters are relatively stable in the SA–DMA nucleation scheme (Olenius et al., 2013, 2017; Ortega et al., 2012; Myllys et al., 2019).

Based on previous studies under atmospheric conditions, the variations in the precursor concentrations, temperature, and CSs do not result in large deviations in the main pathway. Previous simulations under different [SA], [DMA], and temperature have shown that the main pathway was similar under the different conditions studied (Olenius et al., 2013). The effect of CS on the nucleation pathway is dependent on the relative relationship between the coagulation sink and the evaporation rate of a certain cluster. For most clusters out of the specified pathway, the evaporation rates are much higher than the typical CS range in urban atmospheres (Ortega et al., 2012); therefore, such clusters would not dominate the nucleation pathway no matter how low or high the CS is. Thus, in this study, the variation in the dominant pathway under different conditions was ignored.

Accordingly, the parameterization in this study is derived from the nucleation pathway, including A, B, and five other SA–DMA clusters (AmBm (m=1 to 4) and A2B1), consistent with a previous study (Cai et al., 2021b). The clusters, except for A4B4, are assumed to be in pseudo-steady states; i.e., the sink due to evaporation, coagulation scavenging, and cluster collision is equal to the source due to the collisions of molecules or smaller clusters. As the A4B4 clusters are estimated to be with an electrical mobility diameter of approximately 1.4 nm, the pseudo-steady-state formation rate of A4B4 was applied in the parameterization of the J1.4 value in this study. Although some studies have revealed that SA–DMA nucleation could also be enhanced by adding other molecules in certain conditions, a quantitative analysis of these effects in the relevant atmospheric conditions is still lacking; thus in this study, we set up this parameterization based only on the SA–DMA binary nucleation.

The updated parameterization of SA–DMA nucleation was incorporated in the WRF-Chem (Weather Research and Forecasting model with Chemistry). Before adding the SA–DMA nucleation, we already incorporated seven other NPF mechanisms in the model (Zhao et al., 2020), including four inorganic pathways, with binary neutral/ion-induced SA–H2O nucleation and ternary neutral/ion-induced NH3–SA–H2O nucleation, and three organic pathways, including pure organic neutral/ion-induced organic nucleation and ternary nucleation involving organics and SA. The organic-containing nucleation pathways are driven by ultralow- and extremely low-volatility organic compounds (ULVOC and ELVOC), with O : C >0.4 being converted from monoterpene autoxidation. The chemical transformation and volatility distribution of monoterpene is represented in the model by R2D-VBS (radical two-dimensional volatility basis set framework) with constrained parameters against experiments. More details of the R2D-VBS model are given in our previous study (Zhao et al., 2020). The newly formed nanometer-sized particles and their initial size evolution are accounted for in the MOSAIC (MOdel for Simulating Aerosol Interactions and Chemistry) module by 20 size bins covering 1 nm to 10 µm. It is worth mentioning that the newly formed particles from SA–DMA nucleation are lumped into a lower-aerosol size bin in the model than that of other seven pathways. This should be attributed to the fact that our SA–DMA nucleation parameterization is formulated at a 1.4 nm sized particle formation rate, while the remaining ones are fitted based on measured particle formation rates from a cloud chamber at a mobility diameter of 1.7 nm. Given that condensation of gaseous SA and DMA on pre-existing aerosols and nucleation occurs simultaneously in the real atmosphere, we then use a time-integrated-average concentration of the precursors over each time step to drive SA–DMA nucleation in the model. The condensation sink for SA and DMA is calculated according to simulated real-time PNSDs. In addition, the consumption of both SA and DMA concentration during nucleation is also accounted in the model, in order to represent a comprehensive source–sink simulation scheme of two precursors in combination with other settings.

Ambient observations were conducted at an urban site in Beijing from January to April 2018 and from October 2018 to March 2019. The site is located on the West Campus of the Beijing University of Chemical Technology. Details of the observation site can be found in previous studies (Liu et al., 2020; Deng et al., 2020). The concentrations of SA and involving clusters are measured using a chemical ionization high-resolution time-of-flight mass spectrometer (CI-HTOF-MS) and a chemical ionization time-of-flight mass spectrometer with a long mass analyzer (CI-LTOF-MS; Bertram et al., 2011; Jokinen et al., 2012). Other details in the sampling configurations have been reported in our previous study (Deng et al., 2020). Amine concentrations are measured using a modified time-of-flight mass spectrometer (TOF-MS; Zheng et al., 2015b; Cai et al., 2021b). A weather station was deployed to measure the meteorological data, including ambient temperature, relative humidity, and pressure. The PNSDs of particles from 1 nm to 10 µm were measured using a particle size distribution (PSD) and a diethyl glycol scanning mobility particle sizer (DEG-SMPS; Jiang et al., 2011; Liu et al., 2016; Cai et al., 2017a). CS is calculated from the measured PNSDs, and the J1.4 value is calculated using an improved aerosol population balance formula (Cai and Jiang, 2017). The details of instrument calibrations and data validations can be found in our previous study (Cai et al., 2021b).

The reasonability of pseudo-steady-state assumptions in the SA–DMA nucleation pathway was tested through comparisons between the e-folding time of cluster dynamics (τ) in the kinetic simulation (see details in the Supplement) and the time interval of observational data (30 min in this study). The characteristic equilibrium time of involving clusters and simulated J1.4 values are shown in Fig. S1 in the Supplement. Generally, in either clean and cold circumstances or polluted and warm circumstances, the kinetically simulated J1.4 values could be well reproduced by parameterized pseudo-steady-state J1.4 values. Actually, τ values vary greatly with CoagS and γ and would be higher on cleaner and colder days, while even on extremely clean and cold days with CS =0.0001 s-1 and T=255 K, τ of A3B3 (longer than other clusters) is only ∼20 min, which is shorter than the data collection time interval of 30 min. Thus, for circumstances in which there are high atmospheric concentrations of DMA and SA, such as most typical polluted regions, we conclude that nucleation processes are rapid enough that kinetic J1.4 values can be represented by pseudo-steady-state J1.4 values.

Figure 1 presents the comparisons between parameterized J1.4 values in this study and those simulated in the kinetic models (hereafter referred to as KMs) presented by Cai et al. (2021b) and the cluster dynamic simulations containing all AmBn (m, n≤4) clusters (hereafter referred to as CDSs). Generally, there are good consistencies with the simulated J1.4 value between KMs and the parameterization with the correlation coefficient (R2) and normalized mean bias (NMB) of 0.9297 and 0.16, respectively. The simulated J1.4 value in the KMs can be reproduced by a parameterized J1.4 value within a ±50 % range for most of the cases in urban Beijing, with no systematic deviations found between them.

Figure 1b shows that, for most of the circumstances, deviations between the parameterized J1.4 value and the J1.4 value simulated in the CDS are within a range of 1 order of magnitude. The R2 and NMB of the simulated J1.4 value between this parameterization and the CDS are 0.7244 and 0.29, respectively. However, for circumstances with high temperatures, the parameterized J1.4 values were higher than those simulated in CDS, which might be due to the fact that the AkBk (k=2, 3, and 4) clusters are assumed to be non-evaporative in KMs while they would evaporate back in CDSs under high temperatures. The reasonability of the cluster stability assumptions under high temperatures relies mainly on the accuracy of quantum chemistry calculations, which require more experimental evidence and discussions. Additionally, due to the negative dependence of simulated J1.4 values on T, the simulated J1.4 values in this parameterization were mostly lower than 10 cm-3 s-1 under temperatures higher than 15 ∘C, which is lower than the median and mean value of particle formation rates measured during long-term observations in Beijing (Deng et al., 2021). Although they are relatively higher than those simulated in the CDS, the simulation results of the NPF occurrence did not show large deviations.

The computational costs of these three simulations have also been tested on the same personal computer with a MATLAB program. To achieve the steady-state J1.4 values in a specific atmospheric condition, the CDS and KM need ∼10 and ∼0.05 s CPU time, respectively, while the calculation of the parameterized pseudo-steady-state J1.4 value merely costs ∼2×10-7 s CPU time. The CPU time was reduced by a factor of 5×107 and 4×104 when compared to CDS and KM, respectively. Thus, introducing this parameterization into 3-D chemical transport models could greatly reduce the computational costs.

The correlation between parameterized SA–DMA nucleation J1.4 values and the input parameters is shown in Fig. 2. The parameters involved are T, CS, [DMA], and [SAtot]. The mean values of measured data during the observation period (281 K, 0.02 s-1, 3 ppt (parts per trillion), and 3.5×106 cm-3, respectively) are applied as typical conditions in the base case. Different from the semi-empirical power law functions only based on precursor concentrations presented by Dunne et al. (2016), the dependencies of particle formation rates on T and CS are represented in our parameterizations. With T increasing from -10 to 20 ∘C, γ would increase by ∼2 orders of magnitude, as shown in Fig. 2a, and thus the J1.4 value would decrease by over 2 orders of magnitude. This should be attributed to the positive dependence of evaporation rates of A1B1 clusters on the temperature. Under circumstances with high temperatures, the formation of A1B1 and the subsequent formation of larger clusters and 1.4 nm particles would be suppressed. The decreasing trend of the observed NPF rate (J1.5 value in this case) as a function of increasing T in urban Beijing has also been reported (Deng et al., 2020), which is consistent with our parameterizations.

Figure 2b shows that the J1.4 value would decrease by 2–4 orders of magnitude, with CS increasing by a factor of 10, and the logarithm dependence is higher in circumstances with higher CS, such as in urban Beijing, where CoagS dominates the sinks. This is consistent with the negative CS dependence of the measured particle formation rates and NPF occurrence demonstrated in previous observations in Beijing (Deng et al., 2021; Cai et al., 2021a, b). Note that the dependence of parameterized J1.4 values on CS is also sensitive to T due to the synergistic effect of evaporation and coagulation on the sink of A1B1 clusters, which are the key species in SA–DMA nucleation (Cai et al., 2022). If temperatures are higher, then evaporation would be the dominant sink of A1B1 clusters, while CS only suppresses the formation of larger clusters, whereas, under lower temperatures, such as in wintertime Beijing, CS would be the dominant sink of both A1B1 clusters and larger clusters.

The parameterized J1.4 value shows an increasing trend with increasing concentrations of SA and DMA. The parameterized J1.4 value is approximately proportional to [SA]4. This high dependence of the parameterized J1.4 value on [SA] could well reproduce the phenomenon that the rapid formation of new particles usually occurs at noon, which is when there is usually a strong formation of SA molecules in the atmosphere. The dependence of the J1.4 value on [DMA] is decreasing with increasing [DMA]. This is due to the near-saturation formation of A1B1 clusters, which is also found in kinetic model simulation results (Cai et al., 2021b). Generally, the parameterization could reproduce the fact that SA–DMA nucleation is driven by SA–DMA cluster formation, which is dominantly suppressed by cluster evaporation and coagulation sinks.

As with [DMA], [SA] and CS are key input variables for the J1.4 value parameterization, so we first compare the simulated [DMA], [SA], and CS from the DMA1.4_Mech8 scenario with the observations (Fig. 3). Generally, there are good consistencies between both mean values and temporal variations, although there are still deviations at certain times. The mean simulated [DMA], [SA], and CS values are 1.9 ppt, 1.4×106 cm-3, and 0.040 s-1, respectively, which is close to the observed values of 2.0 ppt, 1.6×106 cm-3, and 0.043 s-1. In a quantitative view, the R2 values between the simulated and observed [DMA], [SA], and CS are 0.04, 0.37, and 0.40, respectively, while the coefficients during NPF periods increase to 0.12, 0.51, and 0.49. The normalized mean biases (NMBs) between the simulated and observed [DMA], [SA], and CS are 4.5×10-3, -0.22, and -0.36, respectively, while NMBs during NPF periods are -0.40, 0.01, and -0.66. Generally, the simulation of SA concentrations is good, especially during NPF periods with intense nucleation. We note that the correlation between simulated and observed DMA concentration is lower, which may be attributed to the large uncertainty in the diurnal variation in the amine emission. Nevertheless, during NPF periods, the differences between the observed DMA concentration (0.78±0.60 ppt) and our simulation (1.10±0.60 ppt) is relatively small. For [SA] and CS, to which J1.4 values are most sensitive, we compare the time series of simulated and observed [SA]4 / CS2 (based on the approximate dependence of J1.4 values on [SA] and CS, as shown in Fig. 2) values during NPF periods to show the deviations of the combination of these two input parameters (Fig. S6). Generally, in most nucleation events, the simulated values would not deviate from the observed values by over an order of magnitude. This indicates the validity of the comprehensive representation of input parameters in the model.

The time series of PNSDs for different simulation scenarios are presented in Fig. 4. When SA–DMA nucleation is considered, the typical PNSD shape on observed NPF days (7 December, 8 December, 9 December, 20 January, and 21 January), characterized as the burst of nanometer-sized particles and subsequent growth, is well captured by our best-case scenarios of DMA1.4_Mech8 and also DMA1.7_Mech8. In contrast, the scenarios without DMA-SA nucleation, NoDMA_Mech7 and NoDMA_Mech0, cannot reproduce the observed NPF events with a “vacancy band” for a 1–10 nm size range over the entire simulation period. Actually, although there are slightly higher sub-3 nm particle concentrations in NoDMA_Mech7 than in NoDMA_Mech0, which are generated from the seven nucleation pathways other than DMA–SA nucleation, the newly formed particle concentrations are too low to survive in the subsequent growth and be separated from background aerosols in the PNSDs. These results demonstrate that SA–DMA nucleation should be the dominant mechanism during NPF events in Beijing compared with other seven mechanisms.

Our results also reproduce the dependence of NPF occurrence on CS in Beijing. As shown in Fig. S2 in the Supplement, NPF generally occurs at low CS, while high CS results in nucleation rates that are too low to initiate NPF. The results were also validated through a comparison between the time series of the simulated and observed CS (Fig. 3c). Note that the simulated sub-3 nm particle concentrations also increase slightly on some non-NPF days in the DMA1.4_Mech8 and DMA1.7_Mech8 scenarios; however, the concentrations are ∼1 order of magnitude lower than those on NPF days, and the newly formed particles also fail to survive in the subsequent growth. The improvements of using the nucleation parameterization in this study is further stressed in the comparison between the DMA1.4_Mech8 scenario and the scenario (CLOUD), using the parameterization derived from the experimental data in CLOUD experiments from Dunne et al. (2016). Figure S3 shows that almost no rapid nucleation processes and NPF events are found in the simulation of CLOUD scenarios. In addition to the underestimation of nucleation rates, the simulated high nucleation rates usually occur on observed non-NPF days (Fig. S7), which should be attributed to the ignorance of CS dependence in the power law function parameterizations.

Figure 5 further compares the simulated and observed PNSDs averaged over the simulation period. The best-case scenario of DMA1.4_Mech8 brings the averaged PNSD in the 1–200 nm size range much closer to the observation than those of the base-case NoDMA_Mech7, and the latter only shows a minor change when compared to the scenario NoDMA_Mech0 without any nucleation. One may notice that the averaged PNSD in the 2–10 nm size range for the scenario DMA1.4_Mech8 is still lower than that of observation by ∼1 order of magnitude, despite the good agreement in the number concentrations of particles of ∼1.4 nm. This could be attributed to two possible reasons, namely that the model underestimates the actual nucleation rates or that newly formed particles of ∼1.4 nm grow too fast and reach larger size bins in the model (>10 nm). The first one can be excluded by a generally good agreement between the simulated nucleation rates and the ones derived from an observation, even with a slightly higher mean value for the former (shown in Sect. 3.4; Fig. 6). The observation–simulation comparison of averaged PNSDs is further conducted for individual NPF days. As shown in Fig. S8, the simulated PNSDs on all NPF days follow a similar pattern to the 2-month-averaged one in Fig. 6, indicating that the nucleation in each simulated NPF day is accompanied by subsequent rapid growth. The difference in the concentration of the 2–10 nm particles between observation and simulation is therefore a common feature on various days and is probably attributed to the simplified assumption in the particle growth simulation. Hence, the gap in the 2–10 nm size range might be attributed to the particle growth simulations in the model, which deserves further improvement. Moreover, in spite of a similar performance with respect to improving PNSD simulations compared to the best-case DMA1.4_Mech8, the scenario of DMA1.7_Mech8 presents a PNSD pattern that is shifted to larger size range. For these two scenarios including SA–DMA nucleation, scenario DMA1.4_Mech8 is more reasonable, since a systematic underestimation exists over the entire 1–10 nm range in scenario DMA1.7_Mech8. Still, the conversion from the 1.4 nm rate to those for larger particles through a modified Kerminen–Kulmala equation is an alternative way of depicting the SA–DMA nucleation for other models with different aerosol size settings. Overall, despite the aforementioned deficiencies, our updated WRF-Chem/R2D-VBS model configured with the SA–DMA nucleation parameterization shows substantial improvement in the representation of NPF events and the PNSDs.

Quantitative analysis over various nucleation pathways is performed here to improve the understanding of NPF in Beijing. As presented in Fig. 6, the variation in the nucleation rates, which are derived from observed PNSD data, is well represented by the best-case scenario DMA1.4_Mech8. Compared to the majority contribution from SA–DMA nucleation, the nucleation rates from other nucleation mechanisms are lower by a factor of ∼100. In addition, SA–DMA nucleation contributes over 60 % to the aerosol population, reinforcing its dominant role in modulating the aerosol population in urban atmospheres.

Having shown the significant improvement of the model's performance in simulating the NPF by coupling the SA–DMA nucleation parameterization, we acknowledge that the simulation of the SA–DMA nucleation in 3-D model still has uncertainties in terms of both the source–sink representation of DMA and nucleation parameterization. Here, several key factors which may alter model performance were selected to perform sensitivity analysis.

First, the uncertainties brought by ΔG achieved from different quantum chemistry results are tested for both parameterized J1.4 values and the 3-D chemical transport model simulations. In previous studies, a number of ΔG values have been reported, namely -11.02 kcal mol-1 (Ge et al., 2020), -15.40 kcal mol-1 (Ortega et al., 2012), and -13.54 kcal mol-1 (Myllys et al., 2019). The ΔG of -14.00 kcal mol-1 that was applied in Cai et al. (2021b) to achieve good consistencies between simulated and measured J1.4 values is also applied in the sensitivity analysis. Figure S9 shows the variation in the parameterized J1.4 value by applying different ΔG values at 281 K, which is the median temperature of the observation period. For the DMA with median values of ∼3 ppt, different J1.4 values could vary by ∼5 orders of magnitude, with ΔG between -11.02 and -15.40 kcal mol-1, while J1.4 values with ΔG of -13.54 kcal mol-1 is also lower than that of -15.40 kcal mol-1 by a factor of ∼10. However, if the DMA concentrations are up to ∼30 ppt, then the differences in the J1.4 values, when ΔG varies between -13.54 and -15.40 kcal mol-1, would become much smaller, due to the saturated formation of A1B1 clusters. For the temperature of 298.15 K, the sensitivities of the parameterized J1.4 values are relatively larger because the formation of A1B1 clusters is far from saturation. Generally, the parameterized J1.4 value could be very sensitive to different ΔG values achieved from quantum chemistry results, due to the essential influence of cluster stabilities. As a result, using a lower ΔG value of -15.40 kcal mol-1 in the 3-D simulations with the DMA1.4_Mech8 scenario configuration could lead to much higher nucleation rates compared to the observation (Fig. S10). Thus, we call for a more systematic performance assessment of the quantum chemistry calculation methods to constrain the uncertainties in the cluster thermodynamic stabilities.

Moreover, for the DMA source, we conduct two sensitivity scenarios of doubling (DMA2) and halving (DMA0.5) the inputted DMA emission to test the influence of limited measurements on constraining the DMA / NH3 emission ratio. As for the three sink processes, the parameters for the DMA-⚫OH reaction and wet deposition reported in the literature have relatively small differences, while the aerosol uptake coefficient of DMA covers a wide range that is over 2 orders of magnitude. We then conduct two sensitivity scenarios using the upper (4.4×10-2; Upt4.4E-2) and lower (5.9×10-4; Upt5.9E-4) limit of the aerosol uptake coefficient. All sensitivity scenarios are on the basis of the DMA1.4_Mech8 configuration. The influence of the scaled DMA emissions and varying uptake coefficients on the simulated DMA concentration, PNSDs, and nucleation rate is shown in Figs. S11–S21 in the Supplement. As expected, the DMA concentration, especially for the nighttime spikes, is sensitive to the emission change. In DMA0.5, the simulated J values are lower than those observed in almost all cases. In contrast, although the simulated J values in DMA2 are on average higher than the observations, they are comparable in some specific cases. Considering that during NPF cases, the observed [DMA] values are averagely 1.4 times higher than those simulated in DMA1.4_Mech8, we propose that the slight underestimation of DMA concentrations in this case might be the reason for the underestimation in J values in some cases. The sensitivity analysis for the uptake coefficient, however, shows different results. A higher uptake coefficient of 4.4×10-2 leads to a much lower DMA concentration (10 % of the best case), while the DMA concentration only increases slightly when the lower limit of 5.9×10-4 is used. Moreover, the changes in uptake coefficient show a limited effect on the PNSDs. The reason is that the DMA concentrations during NPF periods are much less affected by the changes in the uptake coefficient than those in non-NPF periods, since NPF usually occurs at low CS conditions when the uptake of DMA is weak.

The sensitivity analysis above shows that the parameters used in our simulation are reasonable, since perturbations within the ranges reported in the literature generally worsen the model performance. We also expect more field measurements of DMA emissions and its aerosol uptake to further constrain the key source–sink process parameters in the simulation of DMA, although some of them show a minor effect on the NPF and PNSD simulations.