We simulated the time evolution of atmospheric cluster concentrations in a one-component system where not only do clusters grow by condensation of monomers, but cluster–cluster collisions also significantly contribute to the growth of the clusters. Our aim was to investigate the consistency of the growth rates of sub-3 nm clusters determined with different methods and the validity of the common approach to use them to estimate particle formation rates. We compared the growth rate corresponding to particle fluxes (FGR), the growth rate derived from the appearance times of clusters (AGR), and the growth rate calculated based on irreversible vapor condensation (CGR). We found that the relation between the different growth rates depends strongly on the external conditions and the properties of the model substance. The difference between the different growth rates was typically highest at the smallest, sub-2 nm sizes. FGR was generally lower than AGR and CGR; at the smallest sizes the difference was often very large, while at sizes larger than 2 nm the growth rates were closer to each other. AGR and CGR were in most cases close to each other at all sizes. The difference between the growth rates was generally lower in conditions where cluster concentrations were high, and evaporation and other losses were thus less significant. Furthermore, our results show that the conventional method used to determine particle formation rates from growth rates may give estimates far from the true values. Thus, care must be taken not only in how the growth rate is determined but also in how it is applied.

Atmospheric new particle formation has been observed to occur frequently in various environments around the world (Kulmala et al., 2004). The process has been estimated to significantly contribute to the global concentrations of cloud condensation nuclei and thus affect the Earth's climate (Spracklen et al., 2008; Merikanto et al., 2009). The primary quantity characterizing new particle formation events is the particle formation rate, which is defined, for any size, as the flux of particles growing past that size (Kulmala et al., 2004). For determining this flux, the particle growth rate (GR) is commonly used (Kulmala et al., 2012).

With respect to analyzing and quantifying new particle formation events, GR
has had several different interpretations and uses. Theoretically, GR for a
given particle is straightforward to define: it is the rate at which the
particle diameter changes at a given moment in time. However, as this growth
is caused by random collisions of vapor molecules, GR can vary a lot in time
and from particle to particle. In particular, all particles of the same size
and chemical composition do not grow at exactly the same rate, as is
inherently assumed in, for example, the condensational growth term in the
standard version of the continuous aerosol general dynamic equation (GDE;
e.g., Seinfeld and Pandis, 2006). Still, a mean size-dependent value can be
derived for GR, resulting in the well-known expressions for the
free-molecular and continuum regimes of condensational growth, as well as
various interpolations for the transition regime (see, e.g., Seinfeld and
Pandis, 2006). These expressions have been used and are convenient when
trying to estimate vapor concentrations from observed GR or vice versa (e.g.,
Dal Maso et al., 2005; Nieminen et al., 2010). In this article, such a growth rate is called the growth rate based on irreversible vapor condensation, abbreviated as CGR (condensational growth rate). Another important use of GR
lies in relating it to the dynamics of the evolving size distribution as the
population of particles undergoes condensational growth. It is used in this
context especially when estimating so-called survival rates, i.e., the
fraction of particles that are not scavenged by background particles,
instrument walls, or other sinks while growing to a certain size. In this
case, it is natural to define GR at a specific diameter by the flux

To study the first steps of new particle formation, the growth rates of
small, sub-3nm particles have been deduced from experimental data using
various methods. The CGR method has been applied to specific measurement
conditions by using the observed concentrations of precursor vapors in the
calculation (Nieminen et al., 2010). The growth rates of charged particles
have been derived from ion spectrometer data by following the time evolution
of the concentration maximum (Hirsikko et al., 2005; Manninen et al., 2009;
Yli-Juuti et al., 2011). Measuring sub-3 nm electrically neutral particles
is challenging, and therefore their growth rates have been indirectly
deduced from the time lag between the rise in sulfuric acid concentration
and the increase in the concentration of 3 nm particles (Weber et al., 1997;
Sihto et al., 2006). However, recent instrumental development has enabled
the detection of neutral clusters with mobility diameters of down to

To investigate the validity of the appearance time method, Lehtipalo et al. (2014) applied the method to particle size distribution data simulated with an aerosol dynamics model. They found that the growth rates determined from the appearance times were close to the average condensational growth rates used as input in the simulation. Olenius et al. (2014) took a different approach to assess the AGR method by using a cluster kinetics model that does not inherently assume any growth rates but simulates the evolution of the cluster population via discrete collisions and evaporations of molecules. They compared the growth rates obtained with the appearance time method (AGR) to the growth rates corresponding to the molecular fluxes (FGR) and concluded that AGR was higher than FGR in the studied conditions. The difference was largest for the smallest clusters and was often strongly affected by the ambient conditions. Although Olenius et al. (2014) showed that AGR and FGR may not be equal, they concentrated on an ideal situation where the growth proceeds only by monomer collisions and evaporations. In reality, there are situations where collisions between two clusters may contribute significantly to the growth (Lehtipalo et al., 2016) and they should, therefore, be taken into account when calculating the flux-equivalent growth rate. Furthermore, Olenius et al. (2014) used a resolution of a single molecule in their analysis, which is not possible when analyzing experimental particle size distributions.

As GR can been interpreted and determined from experimental data in many different ways, it is essential to compare the results obtained with different methods. In this study, we compare the three above mentioned growth rate definitions, FGR, CGR, and AGR, by applying them to modeled particle size distribution data. We use the same dynamic model as Olenius et al. (2014) to simulate the time evolution of cluster concentrations in a one-component system. As opposed to the simulations done by Olenius et al. (2014), in our model system a significant part of clusters' growth proceeds via collisions of small clusters in addition to monomer attachments. Because the growth rate of a single cluster is ambiguous in this case, we group the clusters into size bins for which we calculate the growth rates. This also makes our analysis resemble the analysis of measured particle size distributions with a size resolution poorer than one molecule. We estimate AGR as in Lehtipalo et al. (2014), FGR analogously with Olenius et al. (2014), and CGR directly from the vapor monomer concentration (Nieminen et al., 2010).

Our aim is to answer the following questions: (1) how important are cluster–cluster collisions for the growth of the cluster population in different conditions in our model system? (2) How consistent with each other are the flux-equivalent growth rate, the growth rate derived from the appearance times of the clusters, and the growth rate calculated based on irreversible vapor condensation? (3) How valid is the conventional method used to estimate particle formation rates from growth rates? We examine these questions in different conditions by varying the saturation vapor pressure of the vapor, the vapor source rate and the magnitude of an external sink reducing the vapor and clusters. The simulated conditions correspond to the typical conditions observed during new particle formation in, for example, a boreal forest. In most of the simulations the size-dependent evaporation rate is set to decrease monotonically with increasing cluster size, corresponding to increasing cluster stability. However, we also test a different evaporation profile in order to study the effect of elevated concentrations of stable small clusters on the growth of the cluster population. Furthermore, we investigate how the size resolution, i.e., the width of the size bins, affects the results.

For the growth rate analysis, the clusters were grouped into size bins so
that each bin contains an equal number of cluster sizes, i.e., in linear
volume space the bins are of equal width. The time evolution of the total
cluster concentration

Equation 1 can be obtained directly by integrating the continuous GDE (Friedlander, 1977) for aerosols, including only the
growth and sink terms. If traditional continuous approach is used and
clusters are assumed to grow synchronously by condensation, we can write

In principle, it seems straightforward to combine Eqs. (1) and (2) to obtain
a method to determine growth rates from size distribution data. However, the
possible contribution of larger clusters to growth and the need to somehow
approximate

Here we follow the Eulerian approach used by Olenius et al. (2014) and
referred to as the flux-equivalent growth rate (FGR). The method is based on
defining GR by Eq. (2) even if the original underlying assumptions of Eq. (2) were not valid. Furthermore,

To obtain the net flux

One possible way to assess growth rates from the time evolution of a
particle distribution is based on the times at which concentrations in
different size bins reach their maximum (Lehtinen and Kulmala, 2003). This
is convenient for cases like nucleation bursts where there is a growing mode
of particles. However, in cases where the system approaches a
time-independent steady-state this method obviously does not work. Here we
investigate a method to obtain GR from appearance times of clusters (AGR) in
the size bins, by defining the appearance time

As one of the main reasons to estimate GR from particle size distribution
dynamics is to estimate particle flux (or formation rate)

In addition to the change in the mass diameter of the clusters, we determine
FGR and AGR also as the change in the number of molecules,

The kinetic hard-sphere collision rate between a vapor molecule with
diameter

In addition to the fact that CGR calculated from Eq. (7) takes into account only monomer collisions and no evaporation, the essential difference between CGR and FGR is the perspective from which the growth is studied. CGR corresponds to the traditional Lagrangian approach, where the growth of an individual cluster between different size bins is followed. FGR, however, corresponds to the Eulerian approach, where the net flux between adjacent size bins is studied. See Olenius et al. (2014) for further discussion about the differences between these approaches.

We simulated the time evolution of cluster concentrations in a one-component
system using the Atmospheric Cluster Dynamics Code (ACDC; McGrath et al.,
2012; Olenius et al., 2014). The model included the production of monomers,
all the possible collision and evaporation processes between different
clusters, and the losses of clusters due to an external sink. The model
substance was assumed to consist of spherical molecules and clusters with
the properties of sulfuric acid: a molecular mass (

Summary of the performed simulations.

A summary of the performed simulations is presented in Table 1. In the first
four simulation sets, the Gibbs free energy profile had one maximum and no
minima. In the first simulation set, the effect of monomer concentration was
studied: a constant source of monomer was assumed so that the final
steady-state monomer concentration was 10

In the second simulation set, the aim was to investigate the effect of
cluster stability on the growth of the cluster population by varying the
saturation vapor pressure, to which the evaporation rates are directly
proportional, from 1.5

In the third simulation set, the effect of the magnitude of the external
sink was studied by setting the loss coefficient to 0.7

In the fourth simulation set, we studied how the width of the size bins
affects the growth rates by varying the size bin width from 5 to 14
clusters. Furthermore, as we wanted to compare our results directly with the
results of Olenius et al. (2014) who used an ideal precision of one molecule
in their simulations, we performed additional simulations with a cluster
population that grows only by monomer additions. In these simulations, we
set the saturation vapor pressure of the model substance to 1

The fifth simulation set was otherwise identical with the first simulation
set, but the Gibbs free energy profile was different: a negative term of
90

Finally, we also performed an additional set of simulations by varying the
monomer source rate and the saturation vapor pressure simultaneously. The
monomer source rate was varied between 1

It should be noted that the studied ranges of different parameters, summarized in Table 1, were selected so that our analysis methods were valid in the simulated conditions. If the monomer concentration was set to a too low value, or the saturation vapor pressure and loss factor were too high, the concentration of clusters in the largest size bins would not increase in the simulation and determining growth rates would not be reasonable. On the other hand, if the monomer concentration was very high, or the saturation vapor pressure and the loss coefficient very low, the concentrations of large clusters may become so high that a significant fraction of the flux from a certain size bin would end up not only in the next size bin but also in the size bins larger than that. In this case, the method that we use to calculate the flux-equivalent growth rate would not be valid.

We determined the collision–evaporation fluxes between different size bins
(

Sections 3.1 and 3.2 discuss the results of the simulations where the free energy profile was assumed to have a single maximum and no minima, and the effects of the monomer concentration (Sect. 3.1), the saturation vapor pressure, i.e., cluster stability (Sect. 3.2), and the magnitude of the external sink (Sect. 3.2) on the fluxes and growth rates were studied. Section 3.3 focuses on the effect of the width of the size bins on the results. In Sect. 3.4 the simulations with a different free energy profile, leading to elevated concentrations of small stable clusters, are discussed. Finally, Sect. 3.5 presents the results of the simulations where the monomer source rate and the saturation vapor pressure were simultaneously varied.

The effect of steady-state monomer concentration (shown as
different colors) on quantities describing cluster growth:

In the first simulation set, the steady-state monomer concentration was
varied to see how it affects the growth of the cluster population. When the
monomer source rate, and thus also the steady-state monomer concentration,
increases, the non-monomer fraction of the flux becomes higher (Fig. 1a) as
the relative number of clusters compared to monomers increases. At

In Fig. 1b the true collision–evaporation fluxes from each size bin
(

Figure 1c and d present the different growth rates as a function of the
number of molecules in the cluster and the cluster diameter. FGR is shown as
solid lines, AGR as dashed lines, and CGR as dotted lines. All growth rates
are generally higher when the steady-state monomer concentration is higher.
This is due to higher values of fluxes in the case of FGR and shorter time
between the appearances of adjacent size bins (

We also studied the ratio of AGR to FGR (solid line in Fig. 1e), and
the ratios of CGR to FGR (dotted line in Fig. 1e) and AGR to CGR (Fig. 1f).
AGR is higher than FGR at all sizes, their ratio depending strongly on the
steady-state monomer concentration and the size bin. The AGR to FGR ratio
generally increases with decreasing monomer concentration, reaching the
highest values at

The effect of saturation vapor pressure (shown as different
colors) on quantities describing cluster growth:

In the second simulations set the effect of cluster evaporation rate was
studied by varying the saturation vapor pressure. When the saturation vapor
pressure is lowered from 1

The collision–evaporation fluxes for all size bins (

The flux-equivalent growth rate FGR increases when saturation vapor pressure
is lowered because of larger fluxes (Fig. 2c and d). Except for the
smallest size bin, AGR is also higher with the lower saturation vapor
pressures due to shorter

The ratios of AGR to FGR (solid line in Fig. 2e) and CGR to FGR (dotted line
in Fig. 2e) depend strongly on the saturation vapor pressure and the size
bin. Still, with all three saturation vapor pressures AGR and CGR are higher
than FGR at all sizes. In the smallest size bin, the AGR to FGR ratio varies
between 10

In the third simulation set, the effect of the external sink on the growth
of cluster population was studied by varying the value of the loss
coefficient from 0.7

The effect of size bin width (shown as different colors) on
quantities describing cluster growth:

In the fourth simulation set, the width of size bins was varied too see how the size resolution affects the growth rates. When the size bins are wider, the non-monomer fraction of the flux at a certain size is higher (Fig. 3a). This is partly due to the size dependency of the non-monomer fraction and partly due to the differences in the appearance times of bins with different widths, as the values are determined at the appearance times.

The collision–evaporation fluxes (

The FGR at a certain size, however,
increases when the bin width is decreased (Fig. 3c and d), due to the lower
mean value of the size distribution function of the bin (

The relation of different growth rates to each other is also affected by the
width of the size bins (Fig. 3e and f). The AGR to FGR ratio gets higher
values when the size bins are wider. In the smallest size bin the ratio is
10

In order to compare our results directly with those of Olenius et al. (2014), we performed additional simulations with the saturation vapor
pressure of 1

The effect of steady-state monomer concentration (shown as
different colors) on quantities describing cluster growth in the presence of stable small clusters:

In the fifth simulation set a different cluster free energy profile was used to study the effect of elevated concentrations of stable small clusters on the growth of the population. The contribution of non-monomer collisions to the fluxes between different size bins is significantly increased by the stabilization of small clusters (Fig. 4a, see also Fig. 1a for a comparison). In the smallest size bin the growth mainly proceeds by non-monomer collisions: the non-monomer fraction of the flux is 56–71 % with different monomer concentrations. In the largest size bin, the non-monomer fraction depends strongly on the steady-state monomer concentration: the fraction is 15 % with the lowest monomer concentration and 62 % with the highest monomer concentration.

The collision–evaporation fluxes (

The ratios of AGR to FGR (solid line in Fig. 4e) and CGR to FGR (dotted line
in Fig. 4e) are lower when there are small stable clusters present (see Fig. 1e for a comparison). This is clear especially at small sizes, indicating
that FGR increases there more than AGR or CGR due to the elevated
concentrations of small clusters. The increase of FGR in the smallest size
bin can be explained by a slower decrease of the concentration as a function
of the cluster size in the presence of small stable clusters (see Fig. A4). The AGR to FGR ratio varies between 10

The results of the simulations with different saturation vapor
pressures (

To see the combined effect of external conditions and the properties of
model substance on the growth of clusters, an additional set of simulations
was performed by varying the monomer source rate and saturation vapor
pressure simultaneously. A Gibbs free energy profile containing a maximum
and no minima was assumed. Figure 5a shows the non-monomer fraction of the
flux from the smallest size bin (solid lines) and from the largest size bin
(dashed lines) in all these simulations. The ratio of the monomer source
rate to the loss coefficient (

The ratios of AGR to FGR and AGR to CGR in different simulations are
presented for the smallest size bin (solid lines) and the largest size bin
(dashed lines) in Fig. 5b and c. In the smallest size bin, the AGR to FGR
ratio decreases with increasing

Finally, we also studied the total concentration of clusters (2–70 mers) in
different simulations. Figure 5d shows that the concentration of clusters
increases with increasing

We used a dynamic model to simulate the time evolution of cluster concentrations in a system where cluster–cluster collisions significantly contribute to the growth of clusters. More specifically, we studied how consistent the flux-equivalent growth rate (FGR), the growth rate derived from the appearance times of the clusters (AGR), and the growth rate calculated based on irreversible vapor condensation (CGR) are with each other in different, atmospherically relevant, conditions.

In majority of the simulations the Gibbs free energy of formation of the
clusters was assumed to have a single maximum and no minima, which
corresponds to the increasing stability of clusters with increasing cluster
size. In most of these simulations FGR was lower than AGR and CGR. The
difference was highest, often several orders of magnitude, in the smallest
size bin (at

In one simulation set, a different free energy profile was used, leading to elevated concentrations of stable small clusters, which could correspond to the situation in the atmosphere. In this case, a significantly higher fraction of the growth was due to cluster–cluster collisions than in other simulations. Furthermore, the growth rates of clusters were higher and the different growth rates were closer to each other than in the simulations without stable small clusters.

Moreover, the used size resolution, i.e., the size bin width, was observed to affect the relation between the different growth rates. Generally, the difference between the different growth rates increased with increasing size bin width. Thus, when determining growth rates from measured particle size distributions, size resolution as high as possible should be used.

Altogether, our results demonstrate that different approaches to determine the growth rates of nanometer-sized clusters may give different values depending on the ambient conditions, the properties of the condensing vapor and the clusters, and the size resolution used in the analysis. Especially at the smallest, sub-2nm sizes, the differences between growth rates deduced with different methods can be significant. Our results also indicate that the conventional method used to determine particle formation rates based on growth rates may give estimates far from the true particle fluxes. This should be kept in mind when applying these methods to measured particle size distributions and utilizing the results in particle formation event analyses.

This research was supported by the Academy of Finland Centre of Excellence program (grant no. 272041), the European Research Council (ERC) projects ATM-NUCLE (grant no. 227463), MOCAPAF (grant no. 257360), ATMOGAIN (grant no. 278277), and the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie (grant no. 656994). Edited by: D. Topping