Particle formation rates are usually measured at sizes larger than the critical size at which nucleation occurs. Due to loss of particles during their growth to the detection threshold, the measured formation rate is often substantially lower than the nucleation rate. For this reason a correction needs to be applied in order to determine the nucleation rate from the measured formation rate. Analytical formulae for the correction factor are provided in the literature. However, these methods were derived for atmospheric nucleation measurements and therefore need to be adjusted in order to be applied to chamber nucleation studies. Here we propose an alternative, numerical method that allows precise nucleation rates to be determined in arbitrary experimental environments. The method requires knowledge of the particle size distribution above detection threshold, the particle growth rate, and the particle loss rates as a function of particle size. The effect of self-coagulation, i.e., cluster–cluster collisions, is taken into account in the method.

Aerosol nucleation, or new particle formation (NPF), is an important
phenomenon taking place throughout the Earth's atmosphere (Kulmala et al.,
2004). The key parameter of interest is the nucleation rate, which is defined
as the formation rate (cm

Until recently the smallest mobility diameter that could be measured by a
condensation particle counter (CPC) was 2.5 to 3 nm – which is
substantially larger than the critical size. However, the detection limit of
newly developed CPCs is as small as 1.2 nm in particle mobility diameter
(Sgro and Fernández de la Mora, 2004; Iida et al., 2009; Vanhanen et al.,
2011; Kuang et al., 2012a; Wimmer et al., 2013). Nevertheless, despite this
progress the most widely used CPCs have detection thresholds at 2.5 nm or
above. Moreover, care is needed when interpreting data from the
newly developed CPCs since they can be sensitive to the chemical composition
of the particles (Kangasluoma et al., 2014). Furthermore, CPC cutoff curves
do not have the shape of a step function. Instead, detection of particles
below the cutoff size (usually defined as the size

Kerminen and Kulmala (2002) derived an analytical formula for correcting experimental particle formation rates to determine nucleation rates at a given critical size (abbreviated as the KK method in the following). This method was developed for atmospheric nucleation measurements, and a similar formula was also used by the McMurry group (Weber et al., 1997; McMurry et al., 2005). Several publications followed Kerminen and Kulmala (2002) to include additional effects, like a better description of the coagulation sink from particle size distribution measurements (Lehtinen et al., 2007), self-coagulation (Anttila et al., 2010), and a size-dependent growth rate (Korhonen et al., 2014). In addition to atmospheric measurements, nucleation studies in aerosol chambers or flow reactors have tremendously helped in the understanding of aerosol nucleation. Such experiments require an accurate method to derive the NPF rates. In this study the applicability of the previous methods to chamber experiments such as CLOUD (Cosmics Leaving OUtdoor Droplets) at CERN will be discussed (Kirkby et al., 2011; Almeida et al., 2013; Riccobono et al., 2014). Furthermore, we present here a new method that yields accurate results for any environment – be they chamber or atmospheric data – provided the particle size distribution above a certain threshold size is known, as well as the particle growth rate, and where all loss processes are quantified as a function of size. The new method is verified with the results from a numeric aerosol model.

A lack of suitable instrumentation for the measurement of the particle number
density at diameters below

the only important sink for new particles is their coagulation with larger pre-existing particles,

the new particles grow at a constant rate,

the population of pre-existing particles remains unchanged during the new particle growth.

Coagulation coefficient,

Realizing that the power dependency from Eq. (1) depends on the conditions
during a nucleation event, Lehtinen et al. (2007), in a follow-up publication, dealt with introducing the real power dependency derived from atmospheric
size distribution measurements. This led to the following formulation for the
size correction:

Furthermore, recent findings from atmospheric growth rate measurements
indicate that the GR can be a function of particle size (Kuang et al., 2012b;
Kulmala et al., 2013). Therefore, Korhonen et al. (2014) extended the
analytical solution from Eqs. (4) and (5) and included the effect of a
size-dependent GR, which can either vary linearly with particle size or
according to a power-law dependency. Another effect that can become important
when the population of particles between

The dominant particle loss mechanism for seedless chamber nucleation
experiments is generally due to collisions with the walls of the vessel and
possibly also due to dilution of the chamber gas. Large (3 m) chambers such
as CLOUD have wall loss rates (around 0.001 s

The wall loss rate in chamber experiments can be expressed by (Crump and
Seinfeld, 1981; Metzger et al., 2010)

In addition to wall loss, another mechanism which affects the particle number
density in a chamber experiment is dilution of the chamber gas. Instruments
can take considerable amounts of the chamber gas, and this gas needs to be
replenished in order to maintain a constant pressure. The CLOUD chamber has a
volume of 26.1 m

If coagulation with larger pre-existing aerosols is neglected, which is
well justified in a seedless chamber experiment, the two main loss mechanisms
– wall loss and dilution – can be used to derive an analytical solution for the
NPF rate at a small size. This is achieved by replacing the coagulation loss
term in Eq. (4) from Lehtinen et al. (2007) with

In conclusion, the KK method and also the follow-up versions should only be
applied to chamber nucleation experiments after applying the necessary
adjustments. Equation (10) provides a useful analytical formula for conditions in
which coagulation can be neglected. The data from Fig. 1 provide a guide as to
the relative importance of the different loss mechanisms for the CLOUD
chamber. The wall loss rate for the relevant sizes between 1.4 and 2.9 nm is
on the order of 10

The important conclusion that follows from the comparison of Eqs. (2), (4), and (10) is that experiments and atmospheric environments with similar sink rates cannot be directly compared before corrections are applied, because not only the magnitude of the sink is important but also the dependency of the loss rate as a function of particle size. Despite the practicability of Eq. (10), a new method is required which additionally takes into account coagulation as well as self-coagulation.

We will assume that the size distribution above a certain threshold size
(

The original size distribution above the cutoff size

In order to calculate the formation rate

When applying the method, the particle growth rate GR

In Eq. (15) all quantities are known except the value of

In order to test the relative importance of self-coagulation on the magnitude
of the formation rate correction it is also possible to take into account
only particles at and above

In a recent publication, Olenius et al. (2014) investigated the
relationship between

The accurate definition of

A numerical model was developed recently for the CLOUD chamber to
simulate the formation and growth of uncharged sulfuric acid–dimethylamine
particles (Kürten et al., 2014). The model assumes that particles grow
from monomers by condensation and coagulation. Due to the arguments presented
by Kürten et al. (2014), it has been concluded that
H

The kinetic model is based on McMurry (1980). The time-dependent balance
equation for the monomer concentration

The original model calculated concentrations of clusters ranging from dimer
up to clusters of several thousand molecules. Each size bin was represented
by a single cluster with a fixed number of molecules (or SA

In addition to the kinetic modeling, we have also introduced evaporation rates for the dimer and the trimer (evaporation rates not included in Eqs. 22 and 23 for simplicity). These simulations are used to investigate situations where nucleation and particle growth is dominated by the addition of monomers, because if the evaporation rates for the smallest clusters are sufficiently high, their concentrations become very small and will therefore not contribute significantly to NPF and growth. Although not directly relevant for the sulfuric acid–dimethylamine system, we have calculated the dimer and trimer evaporation rates at 223.15, 248.15, and 278.15 K at 38 % RH from the data presented by Hanson and Lovejoy (2006). Their thermodynamic data were derived for the binary system of sulfuric acid and water. However, the calculated formation rates are not meant to be representative of binary nucleation; rather, they only serve to demonstrate the effect of going from purely kinetic nucleation to nucleation with a relatively large barrier (278 K data). Kinetic nucleation will include collisions with monomers and also show a significant effect from clusters, whereas the new particle formation at 278 K will be dominated by monomer collisions. The other two temperatures show the transition from purely kinetic nucleation to nucleation dominated by monomer additions.

Particle formation rates that have been calculated from the model serve as the reference formation rates to which the reconstructed formation rates can be compared to. We have implemented two separate procedures to calculate the NPF rates, where the first one is following the approach based on Eq. (11) by taking into account all loss processes, while the second one follows the production of particles from two smaller clusters (Eq. 20). The two methods yield exactly the same result, which is a good verification of the kinetic model in this respect.

Figure 4 shows the result of the kinetic model simulation for a monomer
(molecular weight of 143 g mol

Modeled and reconstructed particle size distribution for kinetic
nucleation. The model uses different definitions for the first 100 size bins
(up to

The new universal method to derive a particle formation rate at a smaller
size

The growth rate which is used for the reconstruction is calculated from

A comparison between the accurate solution for the NPF rates and the ones
from the reconstruction method as a function of particle size is shown in
Fig. 5. The accurate solution from the kinetic model is shown by the
solid green line, while the results from the reconstruction method are indicated by
the red triangles. Due to the slight size dependency of the growth rate (it
increases slightly with decreasing size), the reconstructed NPF rates are
somewhat higher than the accurately calculated values. The maximum deviation
occurs at the smallest size and reaches

Formation rates as a function of particle size for kinetic nucleation.
Formation rates simulated with the kinetic model are shown by the green line.
Reconstructed particle formation rates starting at

Using a kinetic model simulation, we show in Fig. 6 an example of the
time-dependent formation rates

Particle formation rates

Using the size distribution as a function of time for particle sizes equal to
or larger than

In the preceding section, the universal method has only been tested for one
sulfuric acid monomer concentration. Variation of the monomer production rate

Formation rates as a function of the sulfuric acid monomer
concentration. The solid blue curves show the formation rates at

In practice, GR will always be subject to measurement uncertainties. In order
to test the sensitivity of the method, the constant GR was multiplied by both a
factor of 1.5 and 0.9. The faster GR leads to an
underestimation in the reconstructed

In order to test the effect of self-coagulation, coagulation has only been
taken into account to occur with particles at and above

The dimer evaporation rate has been set to 2.9 s

Evaporation rates of respectively 181 and 3.1 s

When evaporation rates of respectively 10 060 and 360 s

The Kerminen and Kulmala (2002) method, and its refinements presented in subsequent publications (Lehtinen et al., 2007; Anttila et al., 2010; Korhonen et al., 2014), is widely used in atmospheric and chamber experiments to derive nucleation rates from experimentally measured formation rates at larger particle sizes. However, it was not designed to be applied to chamber nucleation experiments where self-coagulation can be important.

We have therefore presented a new method that yields representative results in any general environment, provided certain quantities are known. The new method requires knowledge of the particle size spectrum above the detection threshold, the particle growth rate, and all loss processes as a function of particle size. With this information the size spectrum and the formation rate can be reconstructed in a stepwise process to a smaller size, where the nucleation rate is determined. The method can give accurate results and, furthermore, takes into account self-coagulation among newly formed particles, which can be an important effect, recognized previously by Anttila et al. (2010). Additionally, if the size-dependent growth rate is available from measurements, it can be readily incorporated during the reconstruction of the size distribution.

The proposed new method allows extrapolation of the particle formation rate
measured at one threshold size (

One general issue with all methods that extrapolate formation rates
towards smaller sizes arises from the uncertainty in the growth rate. In
most cases no measurement of the GR will be available down to the very small
size since the particle number concentrations are also not available
(otherwise no extrapolation of the formation rate would be necessary). A
small size dependency of the GR that is not taken into account can therefore
lead to large uncertainty. In addition, the critical size of the nucleating
particles is generally not known. Ideally, one would choose

Further studies using the new method will focus on the effect of using larger size bins and its application to experimental data measured with condensation particle counters (CPCs) and scanning mobility particle sizer (SMPS) systems.

This research received funding from the EC Seventh Framework Programme (Marie Curie Initial Training Network MC-ITN “CLOUD-TRAIN” no. 316662) and the German Federal Ministry of Education and Research (project no. 01LK1222A). We thank Tinja Olenius and Rick Flagan for helpful discussions. Edited by: H. Grothe