Understanding new particle formation and growth is important because of the
strong impact of these processes on climate and air quality. Measurements to
elucidate the main new particle formation mechanisms are essential; however,
these mechanisms have to be implemented in models to estimate their impact
on the regional and global scale. Parameterizations are computationally
cheap ways of implementing nucleation schemes in models, but they have their
limitations, as they do not necessarily include all relevant parameters.
Process models using sophisticated nucleation schemes can be useful for the
generation of look-up tables in large-scale models or for the analysis of
individual new particle formation events. In addition, some other important
properties can be derived from a process model that implicitly calculates
the evolution of the full aerosol size distribution, e.g., the particle
growth rates. Within this study, a model (SANTIAGO – Sulfuric acid Ammonia
NucleaTIon And GrOwth model) is constructed that simulates new particle
formation starting from the monomer of sulfuric acid up to a particle size
of several hundred nanometers. The smallest sulfuric acid clusters
containing one to four acid molecules and a varying amount of base (ammonia)
are allowed to evaporate in the model, whereas growth beyond the pentamer
(five sulfuric acid molecules) is assumed to be entirely collision-controlled. The
main goal of the present study is to derive appropriate thermodynamic data
needed to calculate the cluster evaporation rates as a function of
temperature. These data are derived numerically from CLOUD (Cosmics Leaving
OUtdoor Droplets) chamber new particle formation rates for neutral sulfuric
acid–water–ammonia nucleation at temperatures between 208 and 292 K. The
numeric methods include an optimization scheme to derive the best estimates
for the thermodynamic data (d

The formation of new aerosol particles from the gas phase (nucleation) is
the most important source of cloud condensation nuclei (CCN) in the free and
upper troposphere (Dunne et al., 2016; Gordon et al., 2017). Binary new
particle formation (NPF) from sulfuric acid and water is thought to be an
important mechanism at cold conditions that can be enhanced by ions (Lee et
al., 2003; Kirkby et al., 2011; Duplissy et al., 2016). The ternary system
involving ammonia besides sulfuric acid and water can yield significantly
enhanced NPF rates (Ball et al., 1999; Benson et al., 2009; Glasoe et al.,
2015; Kirkby et al., 2011; Kürten et al., 2016). The addition of only a
few parts per trillion by volume of ammonia can increase NPF rates by several orders of magnitude
compared with the pure binary system (Kürten et al., 2016). The
importance of ammonia in terms of NPF is highlighted by recent modeling
studies, where a large fraction of CCN originates from ternary

At cold conditions, NPF from

In order to model nucleation, knowledge about cluster evaporation rates is
required. This can either be gained by measurements in a flow tube (Hanson
and Eisele, 2002; Jen et al., 2014, 2016; Hanson et al., 2017)
or in a chamber such as CLOUD (Cosmics Leaving OUtdoor Droplets; Kürten
et al., 2015a). Another possibility is to apply quantum chemical (QC)
calculations (Kurtén et al., 2007; Nadykto and Yu, 2007; Ortega et al.,
2012; Elm et al., 2013; Elm and Kristensen, 2017; Yu et al., 2018).
Comparison between experimental data measured at the CLOUD chamber and
modeled formation rates using the ACDC (Atmospheric Cluster Dynamics Code)
model (McGrath et al., 2012) with evaporation rates from quantum chemistry
(Ortega et al., 2012) yielded good agreement for some conditions (208 and
223 K). For higher temperatures (

For the global modeling studies by Dunne et al. (2016) and Gordon et al. (2017), CLOUD data have been parameterized to yield nucleation rates for four different channels (binary neutral and ion-induced and ternary neutral and ion-induced). The parameterization works well and describes the nucleation rates over a wide range of conditions (Dunne et al., 2016), but it also has its limitations. First, it does not give any insight into the stability of individual sulfuric acid–ammonia clusters. Second, the influence of other parameters on nucleation (e.g., the condensation sink) cannot be tested, while the model by Yu et al. (2018) considers the effect of the condensation sink on the nucleation rate. Third, the parameterization provides only the nucleation rate, while a full nucleation model utilizing size bins over a wide diameter range can also yield the particle growth rates (Li and McMurry, 2018).

In the present study a model covering the aerosol size distribution over a
wide size range, i.e., from the monomer of sulfuric acid up to several
hundred nanometers, is constructed. The model simulates acid–base nucleation
and considers evaporation rates for the clusters containing one to four
sulfuric acid molecules and variable number of base molecules. The model
allows for calculating new particle formation and growth rates at different
sizes and considers sinks like wall loss, dilution and coagulation. SANTIAGO
(Sulfuric acid Ammonia NucleaTIon And GrOwth model) is an extension of a
previous simpler model version used to simulate acid–base nucleation
involving dimethylamine (Kürten et al., 2014, 2018).
The model extension in the present study is a prerequisite for the main goal
of deriving the thermochemical parameters (d

The aim of the present study is to find values for d

The experimental data used to develop the model were taken at the CLOUD
chamber at CERN (European Organization
for Nuclear Research). The 26.1 m

The model used in the present study solves a set of differential equations
describing the concentrations of clusters and particles (McMurry, 1980;
Kürten et al., 2014, 2015a, 2018; McMurry and Li, 2017). The model from
Kürten et al. (2018) describes
nucleation for the system of sulfuric acid and dimethylamine, where the
formed clusters are stable against evaporation at a temperature of 278 K.
For this reason, the sulfuric acid–dimethylamine system can be treated as
quasi-unary, and the kinetic approach (all cluster evaporation rates equal
zero) yields very good agreement between modeled and measured particle
concentrations and formation rates over a wide range of particle diameters.
The model treats the smallest clusters in molecular size bins, based on the
number of sulfuric acid molecules in a cluster, while geometric size bins
are used for the larger particles (Kürten et al., 2018). In the
present study 12 molecular bins and 25 geometric bins with a geometric
growth factor of 1.25 result in a maximum particle diameter of 295 nm.
Choosing a larger number of bins and/or geometric factor would result in a
larger upper size limit, which was, however, not necessary in the present
study. Compared with the earlier study by Kürten et al. (2018) the
number of bins is reduced in order to reduce computation time; the
simulation of one new particle formation event (several hours of nucleation)
takes

While the approach of using a quasi-unary system with zero evaporation
works well for sulfuric acid–dimethylamine, this assumption cannot be used
for sulfuric acid and ammonia because some small clusters evaporate rapidly
(Nadykto and Yu, 2007; Ortega et al., 2012; Jen et al., 2014). In the
following, the number of sulfuric acid molecules denotes the clusters as
monomers (one sulfuric acid), dimers (two sulfuric acids), trimers (three sulfuric
acids), etc. The clusters from the monomer to the tetramer can contain
different numbers of ammonia molecules, where the maximum number of ammonia
molecules is not allowed to exceed the number of acid molecules. The
assumption that no clusters are allowed that contain more base than acid is
based on fast evaporation rates that have been found for such clusters from
quantum chemical calculations (Schobesberger et al., 2015; Elm et al., 2017;
Yu et al., 2018); the assumption is further supported by mass spectrometric
measurements that could not identify such clusters (Kirkby et al., 2011;
Schobesberger et al., 2015). This results in the acid–base reaction scheme
shown in Fig. 1, where

Acid–base scheme implemented in SANTIAGO (Sulfuric acid Ammonia
NucleaTIon And GrOwth model).

While a mixed acid–base cluster can in principle lose either acid or base, the following rule was implemented in the model: clusters containing more acid than base can only evaporate an acid molecule, while clusters containing equal numbers of acid and base can lose a base molecule only. While this is a simplification of the reality, quantum chemical calculations support that this assumption generally considers the dominant evaporation processes (Yu et al., 2018). In principle, acid and base evaporation could be implemented for each cluster in the model, but this would increase the number of free parameters from 22 (with the simplification) to 40 (with all possible evaporations), which would probably not lead to better results but would increase the computation time significantly. The existence of clusters containing more base than acid is excluded in SANTIAGO, which is also supported by quantum chemical calculations (Ortega et al., 2012; Yu et al., 2018).

The thermodynamic parameters for the two smallest pure acid clusters
(

Forward reaction rates are calculated based on the equations for the
collision frequency function by Chan and Mozurkewich (2001), with a value of

The particle growth rate (GR) can be calculated using the monomer and cluster
concentrations in SANTIAGO:

In order to optimize the thermodynamic parameters, it is necessary to define
a criterion that describes the overall deviation between the 125 measured
and modeled new particle formation rates. Since the NPF rates span a large
range (from roughly 10

The optimization method used was introduced by Steihaug (1983) and uses an
approximation for the function,

With the Monte Carlo method (Differential Evolution–Markov Chain algorithm – DE–MC; see Ter Braak, 2006; Ter Braak and Vrugt, 2008; Kupiainen-Määttä, 2016) the pdf's of the thermodynamic parameters are explored. The pdf's give information on the uncertainties of the parameters found by the optimization algorithm, as it is very likely that the optimized values represent a local minimum in the parameter space that is just one possible solution out of many others. The DE–MC algorithm aims at finding the most probable values for the parameters instead of finding the optimal values (Kupiainen-Määttä, 2016). Therefore, the Monte Carlo solutions can be used to evaluate if the optimized values are within the range of the most probable solutions.

At the start of the Monte Carlo simulation, the parameters d

Within the main loop (iterated 5000 times), the first step involves the
random variation of the parameters. The value for each d

In total 20 data sets (each containing 5000 steps) are generated with the
methods described in Sect. 2.5.1 and 2.5.2. From each of the 20 data sets
the average error was determined from the last 2500 points. Whenever the
error for one data set is smaller than the geometric mean from all 20
errors, the data set was selected (Kupiainen-Määttä, 2016). All
selected data sets combined and thinned to 5000 data points represent the
prior distribution,

In the DE–MC algorithm, five Markov
chains are run in parallel, where each of the chains starts from a random
point of the joint history,

The points from the five chains are appended to the joint history,

Probability density functions of d

The results for the thermodynamic parameters are shown in Fig. 2. This
figure indicates the results from the optimization method (dashed lines) and
the pdf's (solid lines) along with their
medians (dotted lines) for the 11 different clusters. A comparison between
the pdf's and the values from Ortega et al. (2012) and Hanson et al. (2017) is
shown in Supplement Fig. S1. The pdf's result from generating histograms of the values
from

Calculated new particle formation (NPF) rates vs. measured NPF
rates (from Kürten et al., 2016). The color code indicates the
temperature (between 208 and 292 K). The calculated values are from the
model using the thermodynamic data from Steihaug's optimization method. The
solid line indicates the one-to-one correspondence, while the dashed lines
indicate a deviation by a factor of 10 from the one-to-one line. The error bars
include the uncertainty of the [

An overall comparison between modeled and measured NPF rates is shown in
Fig. 3. SANTIAGO uses the thermodynamic data from Steihaug's optimization
method. The maximum ratio for the deviation between the modeled and measured
nucleation rates is below a factor of 10, with only a few exceptions. The
average deviation is a factor of

Comparison between simulated and measured new particle formation
rates for five different temperatures. The color code indicates the ammonia
mixing ratio (for the respective temperatures indicated in the figure panels
and a pressure of 1 bar); the grey symbols indicate pure binary conditions.
The model (solid lines) uses thermodynamic data from the optimization scheme
according to Steihaug (1983, Sect. 2.4). The average ratio for the
deviation is

To further evaluate the performance of SANTIAGO the calculated NPF rates are
shown together with the measured rates as a function of the sulfuric acid
concentration for the five different temperatures (Fig. 4). The color code
represents the ammonia mixing ratio, while grey symbols indicate pure binary
nucleation (see Kürten et al., 2016; Duplissy et al., 2016). Again, as
in Fig. 3, the agreement between modeled and measured data is good. The
same applies to the parameterization; in some cases, the parameterization
yields even better agreement compared with the model. This is the case,
for example, for the binary nucleation at 208 K and the data at 278 and 292 K for
the lowest ammonia mixing ratios. However, one clear advantage of SANTIAGO
is that it describes the functional behavior of the system more accurately.
At a temperature of 208 K for the high ammonia mixing ratio, the model line
shows a pronounced curvature, whereas the parameterization yields a straight
line on the log–log plot. The curvature is due to the fact that the survival
probability of subcritical clusters (i.e., clusters below the nonamer) can
be strongly affected by losses to walls or pre-existing particles (Ehrhart
and Curtius, 2013). This effect is most strongly pronounced when the
concentration of the nucleating vapor is relatively low, which results in
slow cluster and/or particle growth rates. Other thermodynamic data sets can be
used to generate model curves similar to the ones in Fig. 4. Using the
data from Ortega et al. (2012) and Hanson et al. (2017) generates Figs. S3
and S4. Figure S2 shows the model curves using d

d

SANTIAGO can yield the dependency of the NPF rates for varying ammonia
concentrations at fixed sulfuric acid concentration. Figure 5 shows these
data for five different temperatures over a wide range of

New particle formation rates as a function of the ammonia concentration. The triangles show the neutral formation rates from the CLOUD experiment normalized to the indicated sulfuric acid concentration for five different temperatures (Kürten et al., 2016). The lines show calculated NPF rates from the model using the thermodynamic data from the optimization method (Table 1). The dashed sections (for 248, 278, and 292 K) indicate regions of the parameter space where the model does not give accurate results, as the true binary rates are expected to be lower (Ehrhart et al., 2016).

For the lowest temperature (208 K) the new particle formation rates show
almost no increase with [

For both temperatures (208 and 223 K) the experimental pure binary new
particle formation rates are represented well by the model. At 248 K and
above, the modeled rates at low [

Figure 6 shows calculated growth rates as a function of the sulfuric acid concentration according to Eq. (2). Additionally, a curve from the equations given by Nieminen et al. (2010) is included. The model results from the present study show a linear increase in the GR as a function of the sulfuric acid monomer concentration as expected (Nieminen et al., 2010). The higher values from SANTIAGO can be explained by the different methods for calculating the collision rate constant that include the van der Waals enhancement for the model of the present study (see Kürten et al., 2018). The increase in GR at a low temperature (208 K) is not intuitive, as the collision rates decrease somewhat with temperature, which should lead to slower GR. However, the clusters are more stable at low temperature and their elevated concentrations can contribute to particle growth (Lehtipalo et al., 2016). This effect is pronounced at 208 K with some ammonia, which indicates that considering only the condensation of monomers is not sufficient for some conditions. Not only can growth be effected by coagulation but also new particle formation rates; therefore, the implementation of a full coagulation scheme (Sect. S1) is important for the present study. The possibility of deriving growth rates with the model is an important feature that is not included in the parameterization for the CLOUD new particle formation rates by Dunne et al. (2016). The modeled growth rates enable further comparison to experimental data and the future study of particle growth to climatically relevant diameters.

Particle growth rates as a function of the sulfuric acid monomer
concentration. The black line indicates the theoretical curve from Nieminen
et al. (2010) for a temperature of 278 K and for sulfuric acid vapor. The
other lines show the calculated particle growth rates at two different
temperatures (indicated in the figure legend). The

The posterior distributions with the median values for d

The distributions for the d

For some clusters, limits seem to exist for d

For most of the clusters, the agreement between the Ortega et al. (2012)
data and the data from the present study is quite good. One exception is
the

One limitation of the model from the present study is that the effect of
water vapor is not taken into account explicitly, i.e., no clusters
containing different amounts of water molecules are considered. However, for
the clusters containing no ammonia, to some extent humidity effects are
included. This is achieved by scaling the evaporation rates of the sulfuric
acid dimer, trimer and tetramer by a factor (20 %/RH)

The effect of water vapor on particle growth rates needs to be studied in
the future. Comparison between measured and modeled growth rates at small
diameters (2 nm) in the acid–base system (sulfuric acid–dimethylamine and
sulfuric acid–ammonia) indicates that water has no significant effect on
particle growth (Lehtipalo et al., 2016). The same can be concluded for the
sulfuric acid–ammonia system at larger diameters (

The fact that no larger clusters than the tetramers can evaporate in
SANTIAGO apparently leads to truncation errors as discussed before for the
binary conditions. This truncation leads to the overestimation of new
particle formation rates for the pure binary conditions at the warm
temperatures. The extent to which truncation affects the ternary new particle
formation can be discussed based on the cluster evaporation rates for the
tetramers at the warmest temperature (292 K). The evaporation rates are

Similarly, to truncation the negligence of evaporation of either acid or base for all considered clusters can potentially lead to errors (see Sect. 2.2). The model includes only the cluster evaporation rates, which seem to be most relevant (see Fig. 1; Ortega et al., 2012; Yu et al., 2018). For each cluster, one evaporation rate is included (either acid or base). This means that the negligence of the second evaporation channel can lead to an overestimation of the cluster concentration. However, in case the omitted evaporation rate is smaller than the considered one, this effect is very likely small. The selection of the considered evaporation rates are guided by the literature data on QC calculations (Ortega et al., 2012; Yu et al., 2018). This does, however, not rule out that important evaporation channels could be neglected. On the other hand, increasing the number of free parameters does not necessarily improve the accuracy of the model but only its complexity and the computational demands for the optimization and Monte Carlo calculations.

The previous study by Kürten et al. (2016) compares the CLOUD data with ACDC (McGrath et al., 2012) model calculations using the thermodynamic data from Ortega et al. (2012). Using the same data, Fig. S3 shows this comparison using the model from the present study. Surprisingly the agreement between the model and measurement is better than in the study by Kürten et al. (2016). One difference between the two studies is that the ACDC model used the formation rate for neutral clusters containing six sulfuric acid molecules instead of nine in the present study. This difference was tested with the present model, but it only leads to a very small change in the simulated formation rates. An effect that can, however, explain the discrepancy is that the ACDC model calculations did not consider a wide range of particle sizes. This could lead to inaccuracies regarding the coagulation sink for the formed clusters. Especially at high acid concentrations when growth and nucleation rates are large, the particles can create a significant sink that can reach a similar magnitude to the wall loss rate in the CLOUD chamber (Kürten et al., 2015b). Neglecting the full size distribution can lead to an overestimation of cluster concentrations and formation rates (Sect. S1). This effect needs to be studied in more detail in the future. In any case, taking into account particles over a wide size range should improve the accuracy of a model due to the described effect.

The comparison between the CLOUD data and SANTIAGO using the Hanson et
al. (2017) data is shown in Fig. S4. Hanson et al. (2017)
base their data on flow
tube measurements performed at rather warm temperatures (

The model SANTIAGO (Sulfuric acid Ammonia NucleaTIon And GrOwth model)
describes new particle formation and growth from the reactions between
sulfuric acid and ammonia. The effect of water vapor is taken into account,
but the capability of simulating binary nucleation is limited to low
temperatures (

SANTIAGO implements evaporation of the smallest clusters, containing one to
four sulfuric acid molecules and a variable number of ammonia molecules. The
thermodynamic data (d

The optimization and the Monte Carlo method were successfully applied to
explore the landscape of the cluster thermodynamics for the nucleating
system of sulfuric acid and ammonia. However, the probability density
functions from the DE–MC algorithm do not yield a very clear picture of the
most likely values for d

Implementation of the literature data in the model indicates that the Ortega
et al. (2012) thermodynamic data describes the CLOUD data better than
previously thought (Kürten et al., 2016). This could be because of the
negligence of large particles in the previous study. It seems essential to
include the larger nucleated particles in the model, as these contribute to
the sink for the small nucleating clusters and particles. The Hanson et
al. (2017) data overestimate the new particle formation rates for the warm
temperatures (278 and 292 K). No direct comparison to the Yu et al. (2018)
is possible, as no temperature-dependent evaporation rates can be calculated
from their reported d

SANTIAGO allows for calculating new particle formation rates for a wide range of
experimental conditions (

Finally, the strong dependence on [

Data can be downloaded from

The supplement related to this article is available online at:

The author declares that there is no conflict of interest.

This article is part of the special issue “The CERN CLOUD experiment (ACP/AMT inter-journal SI)”. It is not associated with a conference.

Funding from the German Federal Ministry of Education and Research (project “CLOUD-16” 01LK1601A) is gratefully acknowledged.

This paper was edited by Jonathan Abbatt and reviewed by two anonymous referees.