Impacts of coagulation on the appearance time method for new particle growth rate evaluation and their corrections

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 sizedependent 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 %.


Figure S1
The growth rate estimated using the appearance time method for particle growth with a volatile vapor and a nonvolatile vapor. The concentration of these two vapors, N1 and N2, are assumed equal and constant. β is the coagulation coefficient between the vapor and a particle and it is a function of the particle size. The evaporation rate of the volatile vapor, 20 E2, is assumed to be size-dependent. As indicated in the figure, the critical size for the volatile vapor is ~2.5 nm. The evolution of aerosol size distribution is simulated using a 2-dimentional discrete model. Particle coagulation is neglected in this simulation. The theoretical value of net condensation is obtained based on particle growth flux.

Figure S2
The appearance time method under a varying vapor concentration. The test condition is similar to that of Fig. 7, as summarized in Table A1, No. 8, and the only difference between these two tests is the size of the vapor molecule. The relative molecular masses of the vapor molecules for this test and the test in Fig. 7 are assumed to be 400 and 143, respectively.
As a result, the shapes of the growth rate curves in this figure are similar to those in Fig. 7 but they are shifted towards larger 30 particle sizes and higher growth rates.
Page 3 of 5 Figure S3 Influence of the variation of coagulation sink on the growth rate estimated using the corrected appearance time 35 method. a) Condensation sink contributed by background particles and the normalized particle concentrations as a function of time. The background condensation sink (CSbg) is contributed by a certain concentration of 100 nm particles, which varies with time during the growth of 1.2-3 nm particles. CSbg characterizes the coagulation sink (CoagS) of particles contributed by these 100-nm background particles. The particle concentration is normalized by dividing its steady-state concentration.
Due to the decrease of CSbg, the maximum value of the normalized particle concentration exceeds 1.0. b) Theoretical particle 40 growth rate and the growth rates estimated using the conventional and corrected appearance time method. The theoretical growth rate is calculated using the vapor condensation rate. The 50% appearance time is calculated using the steady-state concentrations. The coagulation sink contributed by both the 100-nm particles and new particles are accounted for when correcting the influence of coagulation sink. The deviation between theoretical growth rate and the estimated growth rate using the corrected appearance time method is mainly caused by variation of coagulation sink, which is not accounted for in 45 the correction formula.

The impacts of coagulation source and their corrections
In this section, we present a derivation for Eq. 6. 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 60 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. S1) 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; and Ni is the concentration (cm -3 ) of particle i. N1 is assumed to be 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: 65 where Src is the 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 Ni-1,∞.
Page 5 of 5 Now we consider the scenario with coagulation sink and coagulation source (Eq. 19 where CoagSi is the coagulation sink (s -1 ) corresponding to Ni; and CoagSrci is the coagulation source term (cm -3 ·s -1 ) 70 corresponding to Ni.
As illustrated in the main text, CoagSi and CoagSrci change both Src and Ni,∞. The appearance time and hence the retrieved growth rate are mainly influenced in two aspects: 1) the steady-state concentration and 2) the particle source that determines the time to reach a certain steady-state concentration. The conventional (apparent) growth rate under this scenario can be obtained by approximately accounting for these two aspects, i.e.,