Atmospheric new particle formation occurs frequently in the global atmosphere and may play a crucial role in climate by affecting cloud properties. The relevance of newly formed nanoparticles depends largely on the dynamics governing their initial formation and growth to sizes where they become important for cloud microphysics. One key to the proper understanding of nanoparticle effects on climate is therefore hidden in the growth mechanisms. In this study we have developed and successfully tested two independent methods based on the aerosol general dynamics equation, allowing detailed retrieval of time- and size-dependent nanoparticle growth rates. Both methods were used to analyze particle formation from two different biogenic precursor vapors in controlled chamber experiments. Our results suggest that growth rates below 10 nm show much more variation than is currently thought and pin down the decisive size range of growth at around 5 nm where in-depth studies of physical and chemical particle properties are needed.

Aerosol nanoparticle formation from gas-to-particle conversion occurs
frequently throughout the global atmosphere

Importantly, the formation rate at a specific diameter

Schematic comparison of defined regions

Change rates of the number-size distribution are described by the continuous GDE as in

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

The first method in estimating particle growth rates is based on the
assumption that regions

The particle number concentration within each region

This equation can be solved for the simulated
(

Growth is decoupled from the other dynamic processes. Thus large relative
changes in the limits of the region

Rapid changes in the growth rate require adequate time resolution of the experimental data as the result of the analysis method being a mean growth rate for the respective time interval.

Influence of the coagulation process by particles smaller than

The present method utilizes integral values to determine the growth rate. Thus local minima and maxima of the measured number-size distribution (e.g., due to low particle concentration) may cancel out. However, this depends on the choice of the width of regions, which can be set for each analysis run.

The second method is based on explicit manipulation of the adapted,
size-integrated GDE (see Eq.

Results of the two GDE-based analysis methods on simulated input
size distribution generated by the SALSA model. Panel

The first term on the right-hand side considers particles that grow into the
range [

The INSIDE method also features aspects 1 to 3 of the TREND method.

It allows for determination of GR at pre-selected diameters
while TREND method determines GR and

Fluctuations or scatter in the input number-size distribution may significantly
change the result due to the

In order to test the analysis methods described above, number size
distributions generated by the model SALSA (Sectional Aerosol module for
Large Scale Applications;

Combined, DMA train and SMPS data, showing size distribution
evolution over time for the ozonolysis of two different VOCs.
Panel

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.

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 m

Figure

These different dynamics can be quantified by analyzing the evolution of the
number-size distribution with the two methods described above. In
Fig.

Growth rate analysis of the

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

Additionally, the results from the TREND method are compared with growth rate
values calculated by the appearance time method in
Fig.

This seems to be completely different in the

Growth rate analysis of the

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

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

Despite the challenges in the highly dynamic case of

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

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;

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

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

The main input data and results of this publication can be
found under

The flowchart contained in Fig.

The input (experimental or simulated) number-size distribution (at time

Flow chart describing the principle of the data analysis.

Figure

The result of the aerosol dynamics module is a simulated distribution

Wall loss of particles is described by

Dilution is described similarly to wall loss. For the description of dilution
we assume that the chamber is well mixed (i.e., no concentration gradients,
which has been verified in the NCAR chamber using CO

The result of the simulation is a particle size distribution

Flow diagram of the aerosol dynamics model calculating the changes
to a particle size distribution

In order to test the models' performance when coagulation must be
considered, similar simulations featuring higher nucleation rates (

Analyzed (straight lines) and set growth rate (dashed lines) as a function of
particle diameter. Panel

Figure

We therefore conclude that our methods can handle the effects of coagulation, and the small discrepancies of this test analysis are due to different simulation representations, which will not occur when experimental data sets are used. However, in the cases of low particle growth rates and high coagulation sinks, the effects of coagulation might become more important and a more detailed quantification of coagulation effects might be necessary.

As described in Sect.

In order to estimate the effect of particle coagulation of sub-detectable
sizes on the growth rate, we first generate a particle size distribution
based on

particle density of 0.5 to 2 g cm

monomer volume of 0.2 to 0.8 nm

monomer formation rate of

The resulting number-size distributions are depicted by
Fig.

Effects of sub-

Size distributions generated by the SALSA module and used for the
testing of the analysis methods in Sect.

The number-size distribution in the diameter range from 0.8 to 3 nm was
divided in five logarithmically spaced sections. In the next step the
contribution to the growth rate per hour of a particle due to coagulation
with particles (constant concentration of 1 particle cm

Multiplication of this growth rate function by the number-size distribution
as determined above results in the growth rate (nm h

For the experimental data obtained at the NCAR biogenic aerosol chamber a
combined data inversion procedure for the DMA train and the SMPS measuring
the same aerosol source is applied. Both instruments rely on electrical
mobility analysis done by differential mobility analyzers (DMAs). While the
SMPS operates one DMA in scanning mode, the DMA train operates five DMAs
in parallel at fixed voltages and hence particle sizes. Data inversion is
based on the procedure of

Note that for both instruments most of the parameters are distinct. Sampling
efficiencies are inferred from sampling line lengths, sample flow rates and
classified diameters and assumed to follow the diffusional losses according
to

Moreover, Eq. (

In both of the measurements presented above, the SMPS measured down to
10

Furthermore, the DMA train does not rely on a scanning procedure and
therefore acquires concentration data at the fixed sizes within each second.
The SMPS requires 120 s to scan from low voltage to high voltage and
another 120 s to reverse. The results from each

In the following potential errors of the analysis methods TREND and INSIDE are discussed. Note that errors originating from the experiment are not part of this section, which solely describes the error caused by the analysis methods itself. Both analysis methods are not exact as they are derived from quantities that are either averaged (with respect to time and/or particle diameter) or generated by means of numerical simulation.

Further, both methods rely on simulated particle size distributions. In case coagulation is not dominant (as it is the case in the present work) the error due to numerical simulation can be neglected. Other simulation errors may originate from dilution of the aerosol and particle wall losses. Given that these processes are known (i.e., determined experimentally), the simulation result is on the order of 0.1 %.

An additional source of uncertainty is the fact that particle growth cannot
be taken into account for the calculation of other dynamic processes since it
is determined from the simulated data. This affects coagulation and wall
losses. In order to estimate the effect of particle growth on the calculation
of wall loss, the change in median diameter determined by the TREND method is
considered. For

Considering inter-particle forces

The mean resulting error and corresponding standard deviation are 0.4 and
7.6 %, respectively, for the

The TREND method calculates growth rates for

To conclude, for experiments with

Input data used to generate particle size
distributions with the SALSA module.

PMW, JO, TK, PHM and JNS performed the experiments; LP, DS, PMW and PHM developed the analysis tools; DS and LP analyzed the data; HK, AL and KEJL provided the simulation input; LP, DS, PHM, KEJL and PMW were involved in the scientific interpretation and discussion; and LP, DS and PMW wrote the manuscript.

The authors declare that they have no conflict of interest.

This work was supported by the European Research Council under the European Community's Seventh Framework Programme (FP7/2007–2013)/ERC grant agreement no. 616075. Peter H. McMurry's work was supported by U.S. DOE grant DE-SC0011780. James N. Smith acknowledges funding from DOE under grant no. DE-SC0014469. The National Center for Atmospheric Research is supported by the National Science Foundation.Edited by: Hang Su Reviewed by: two anonymous referees