Aerosol nanoparticle formation from gas-to-particle conversion occurs frequently throughout the global atmosphere . Despite their small sizes these particles might be of climate relevance through the indirect aerosol–cloud effect . Modeling results suggest that this secondary aerosol formation mechanism contributes roughly 50 % of particles to the budget of cloud condensation nuclei . New particle formation (NPF) has been the subject of numerous studies for several decades. Besides experimental studies under ambient and laboratory conditions, substantial effort has been put into the modeling of aerosol dynamics to address phenomena such as nucleation, condensation/evaporation and coagulation. In order for newly formed particles to eventually become cloud condensation nuclei, particles need to grow sufficiently fast to prevent them from being scavenged by pre-existing particles.

Importantly, the formation rate at a specific diameter Jdp is highly sensitive to the diameter growth rate ddp/dt. Knowledge of ddp/dt is needed not only to calculate particle formation rates; it intrinsically contains information on the growth mechanisms . The diameter growth rate as a function of particle size and time is therefore key to the understanding of growth mechanisms during gas-to-particle conversion. Several authors have characterized growth rates from the first appearance of various particle sizes over time, which is referred to as the appearance time method e.g.,. Others showed that a fit of a lognormal distribution to the nucleation mode particles can be used to infer the growth of the population over time . However, these methods cannot fully resolve both the size and time dependencies of the observed growth rates in highly dynamic systems. Therefore, several attempts have been made in the past to derive ddp/dt by solving the general aerosol dynamics equation (GDE) using growth rate analysis on the basis of experimental number-size distribution measurements . Those techniques typically require some kind of fitting and additionally might suffer from insufficient data quality, which is usually limited over a certain size range and/or time resolution of the sizing technique being applied. Number-size distribution measurements typically take 1–2 min per scan, and can therefore be too slow to characterize the observed size-resolved growth rates. Hence, determination of ddp/dt is still a major source of uncertainty in the proper characterization of nanoparticle growth. Here we present a new approach to this problem that compares two different methods based on GDE analysis. The methods are tested and compared to simulated NPF events. Both approaches are then applied to experimental data from particles formed from the ozonolysis of monoterpenes and sesquiterpenes in a 10 m3 aerosol chamber. State-of-the-art particle sizing instrumentation enables the methods to quantify size- and time-dependent growth rates over the crucial size range between 2 and 50 nm. This thereby helps to improve our understanding of the differences between monoterpenes and sesquiterpenes in new particle formation, which have been emphasized by recent studies .

Change rates of the number-size distribution are described by the continuous GDE as in : ∂n(v,t)∂t=12∫0vK(v-q,q)n(v-q,t)n(q,t)dq-n(v,t)∫0∞K(q,v)n(q,t)dq-∂∂v[I(v)n(v,t)]+S(v)-R(v), where n(v,t) represents the number volume distribution at time t and volume v, K(v,q) is the coagulation kernel, I(v) a particle current across the volume v and S(v) and R(v) are size-dependent source or removal terms, respectively.

In a well-controlled aerosol chamber experiment, the GDE is governed by just a few effects. An aerosol dynamics module accounting for dilution, wall losses and coagulation is used to calculate simulated number-size distributions nsimtj+1,dp evolving from measured experimental input number-size distributions nexptj,dp between two subsequent time steps tj and tj+1. With growth as the only unknown in the GDE, comparison between the simulated and the measured number-size distribution nexptj+1,dp allows for its quantification. Here we focus primarily on two different methods that have been employed to determine size- and time-dependent growth rates from this comparison. A brief description of the software tool used to interpret the experimental data is given below; details can be found in Appendices  and . Potential errors of the analysis methods are discussed in Appendix .

In order to test the analysis methods described above, number size distributions generated by the model SALSA (Sectional Aerosol module for Large Scale Applications; ) were used. Detailed information about the input parameters for the SALSA model can be found in Appendix . Figure  shows the growth rate functions serving as input for the SALSA model and the results of the two analysis methods.

Note that no fitting was done. Both models capture the slope of the input growth rate curve well; however, there are some deviations. Both models show an increasing scatter of the data with increasing particle diameter. This can be explained by the different representations of the number-size distribution within the models. While the SALSA model uses a volume-based moving average representation, the analysis methods consider a distribution of the particles within each size bin. Thus the larger the particles grow the more pronounced the differences between the set growth rate (applied by SALSA) and the analyzed growth rate (determined by INSIDE and TREND) become. Furthermore, some pronounced deviations between measured and determined growth rate occur for the INSIDE method, which are not found for the TREND method. They show up only at the upper end of the number-size distribution where number concentrations are low.

These pronounced deviations are not found in the TREND method which uses integrated number concentration values with respect to dynamic diameters (see Fig. ) to determine growth rates and hence all regions have fixed counting statistics. Except for this problem, both methods are able to determine growth curves by analyzing dNdlog⁡dp representations of number-size distributions especially for small particle sizes and low particle numbers. Statistical analysis of the deviation between the generated growth rate (SALSA) and results from the INSIDE and TREND method reflect this behavior. Each analyzed growth rate data point is compared to the SALSA input value at the same diameter. The mean relative deviation and the corresponding standard deviation are 1.2 and 5.0 % for the TREND method and 6.5 and 12.7 % for the INSIDE method, respectively. The effect of higher particle concentration and hence larger influence of coagulation has been investigated in similar simulations featuring higher nucleation rates and are discussed in Appendix . It seems that the TREND method works better for analyzing the leading edge of the newly formed particle size distributions, and it in general shows less scatter due to its integral method. On the other hand, the INSIDE method performed very well when analyzing GR at higher particle concentrations; however, it is more sensitive to scatter in the experimental input data which has to be considered when real data is analyzed. Note that statistical errors occurring especially during the measurement of low concentration aerosols may cause deviations between observed and analyzed growth rates.

Both methods described above were used to analyze growth rates from NPF events produced in the aerosol chamber at the National Center for Atmospheric Research (NCAR) in Boulder, CO, USA. Experiments were performed in a 10 m3 teflon bag which was continuously flushed by zero air at a flow rate of 40 L min-1. A biogenic volatile organic compound (VOC; α-pinene or β-caryophyllene) was added to the zero air until steady state concentrations of ∼ 4 ppb were obtained. Subsequently, a UV mercury lamp was turned on in one of the zero air lines to increase ozone in the chamber steadily and initiate ozonolysis of the VOC and subsequent NPF. For both experiments steady-state ozone concentrations of ∼ 25–30 ppb were reached at the end of the experiment. Additional details of the experimental setup can be found in . Evolution of the number-size distribution was monitored by a regular scanning mobility particle spectrometer (SMPS) and a prototype differential mobility analyzer train (DMA train). The DMA train uses several DMAs and condensation particle counters (CPCs) in parallel. Each DMA is set to transmit only particles of a specific mobility diameter to monitor the size evolution of individual sizes preferably in the sub-10 nm size range at high time resolution (on the order of seconds). Operation principles from a similar, advanced setup can be found in .

Figure  shows combined size distribution measurements for particles from both VOCs. Details of the data inversion procedure can be found in Appendix . It can be clearly seen that not only the absolute particle yield is higher in the β-caryophyllene system, but also growth proceeds much faster than in the case of α-pinene. While the first appearance of particles is observed after ∼ 10 min for the sesquiterpene, it takes roughly 3 times as long for the monoterpene. Obviously, there is quite different growth dynamics involved.

These different dynamics can be quantified by analyzing the evolution of the number-size distribution with the two methods described above. In Fig.  the results for the α-pinene system are shown. Both methods show the same trend and similar absolute growth rate values. As already discussed during the test with simulated size distributions, the results of the TREND method do not cover the full size range at every time step due to the choice of the size interval number m. The INSIDE method on the other hand generally shows more scatter, especially in regions where counting statistics above the evaluation size deval are poor and therefore those results are greyed out.

This analysis reveals that growth rates above 10 nm have a negligible size dependence. However, a strong size dependence is seen below 10 nm with peak growth rates around 7 nm and strongly decreasing growth rates in the sub-5 nm size range, independent of the measurement time. This can be explained by a multi-component Kelvin effect, where some of the α-pinene reaction products can only participate in growth when particles have grown large enough to overcome the Kelvin barrier, as shown in for the α-pinene system. For the peak at 5 nm we can exclude the contribution of particle coagulation below the measurement size range as shown in Appendix .

Additionally, the results from the TREND method are compared with growth rate values calculated by the appearance time method in Fig. c, as this method is used in other chamber studies . The appearance time method shows a similar trend as TREND method for the three possible appearance time measurements. The appearance time method does not reveal the complete time and size dependencies of the growth, and could neither conclude about the observation of a multi-component Kelvin effect nor about the observed higher growth rates at around 5 nm. Moreover the TREND method shows the clear trend of increasing growth rates until a more or less steady state growth is reached. We speculate that this is due to the slow accumulation of condensable low-volatility vapors by the ozonolysis (proceeding at a rate constant of kαp.×O3 (293 K) =9.06×10-17 cm3 molecule-1 s-1; ) and subsequent auto-oxidation of α-pinene.

This seems to be completely different in the β-caryophyllene system. The results of the growth rate analysis are presented in Fig. . The corresponding comparison of the results from the TREND method with the calculated growth rates from the appearance time method and with growth rates inferred by the lognormal distribution method are displayed in Fig. c and both show good agreement. The lognormal distribution function method was not applied to the α-pinene data set as measured particle size distributions differ strongly from lognormal shape.

High growth rates at the beginning of the observed events are followed by a drop of growth rates in all sizes as the particle growth goes on. This can be explained by the very high oxidation potential and high reaction rates of β-caryophyllene, where the reaction rate constant for ozonolysis (kβc.×O3 (298 K) =1.2×10-14 cm3 molecule-1 s-1 ) is three orders of magnitude higher than in the case of α-pinene. Condensable vapors are therefore quickly formed and the steady state β-caryophyllene concentration at the beginning of the experiment is depleted by this fast reaction. Together with the fast build up of a large condensational sink, this shuts off new particle formation and reduces the fast growth rates.

In such a highly dynamic case it becomes evident that time resolution greater than the 240 s from the SMPS scans would yield a better data set for the applied analysis methods. Additionally, when the particles reach larger sizes, the higher total particle number concentration increases the influence of coagulation and might disturb the results derived at small sizes. Moreover, due to the higher particle number concentrations in the growing mode, the inferred size range of the growth rates by the TREND method shrinks. A more detailed discussion of the uncertainties of the two methods can be found in Appendix . In general, the biggest sources of errors are low time resolution of the measurement data and scattering of the experimental data. Further, a source of high potential error is coagulation. For the experimental data presented in this work, the estimated error of the GR determination associated with the analysis tools is typically in the range from 2 to 35 % depending on the analysis method and the experiment.

Despite the challenges in the highly dynamic case of β-caryophyllene ozonolysis, both methods reveal that the size dependence of the growth rates is most significant in the sub-10 nm region, as in the case of α-pinene. Moreover, the INSIDE method still covers the full size range for the analysis of the size dependence. When new particles are formed at the beginning of the experiment it reveals extremely high growth rates of up to 250 nm h-1 between 5 and 10 nm. Similar to the case of α-pinene, but somewhat less significant, are the lower growth rates in the sub-3 nm range. The smaller reduction of growth in that size range indicates that the vapors produced from the ozonolysis of β-caryophyllene are less volatile compared to the products of α-pinene and can therefore participate in growth starting from the smallest sizes and greater. This is predictable because a sesquiterpene with 15 carbon atoms will be less volatile than a monoterpene e.g.,.

We presented two methods to determine size- and time-dependent growth rates by analyzing particle size distributions and solving the GDE. The TREND method tracks regions of the number-size distribution. The INSIDE method is based on interpreting the size-integrated GDE, and determines growth rates at certain deval.

Both methods reliably reproduce input growth rates from simulated size distributions and allow for quantitative comparison. The TREND method generally shows less scatter and less sensitivity to low counting statistics but cannot always cover the full range of particle sizes where growth is actually observed. The INSIDE method is capable of determining growth rates wherever particles are measured. However, determination of growth rates at very low or very high particle concentrations may suffer from considerable errors. This is due to insufficient counting statistics of the measured input data on the one hand, and considerable coagulation effects on the other hand. While coagulation is typically considered in the GDE analysis a precise description of coagulation requires detailed knowledge of the aerosol properties (e.g., inter-particle forces or shape; ), which are typically unknown for newly formed particles. Moreover, coagulation will become more important when ambient data with high background aerosol loadings are considered. Theoretically, both methods could directly incorporate a background aerosol as long as it can be separated from the nucleation mode. A more detailed implementation of background aerosol and coagulation effect will be necessary for the application of the methods to ambient data sets.

We applied our methods to experimental size distribution data from chamber studies to derive size- and time-dependent growth rates from ozonolysis of two different biogenic VOC precursors. Both methods agree well with the widely used appearance time method and provide valuable insights on some unexpected details of the growth dynamics in these systems.

For both studied VOC systems, a strong increase in growth rates was found for the smallest diameters until a maximum value was reached at around 7 nm. This finding strongly suggests that (biogenic) growth is governed by a multi-component Kelvin effect which allows for condensation of vapor molecules only if the particles exceed a certain size. This observation is very pronounced in the case of a-pinene and was reported independently from other studies . For the β-caryophyllene system, it is less significant, indicating that the majority of β-caryophyllene ozonolysis products are generally less volatile and can participate in growth at particle diameters well below 10 nm. This system showed highly dynamic behavior and fast changing growth rates over time, as the condensable vapors quickly became depleted in the chamber due to the high reactivity of β-caryophyllene and low-volatility reaction products. Growth rates above 10 nm generally showed only minor size dependence. Regarding the different nanoparticle-forming behavior of monoterpenes and sesquiterpenes, similar findings were recently reported from plant emission studies in a chamber environment .

Our analysis underline the critical need to accurately quantify growth dynamics in the sub-10 nm size range. This range is crucially important for the survival probability of newly formed particles and clearly features the biggest changes in growth rates. One of the prerequisites for the successful application of our newly developed methods is having size distribution measurements providing time resolution below 1 min and good counting statistics. We see these requirements fulfilled in latest state-of-the-art instrumentation , allowing full exploitation of growth dynamics in the future. We also plan to make the analysis tool kit publicly available in order to allow for wide application and improvement by the scientific community.