Evaporation rates of small negatively charged sulfuric
acid–ammonia clusters are determined by combining detailed cluster formation
simulations with cluster distributions measured in the CLOUD experiment at CERN. The analysis is
performed by varying the evaporation rates with Markov chain Monte Carlo
(MCMC), running cluster formation simulations with each new set of
evaporation rates and comparing the obtained cluster distributions to the
measurements. In a second set of simulations, the fragmentation of clusters
in the mass spectrometer due to energetic collisions is studied by treating
also the fragmentation probabilities as unknown parameters and varying them
with MCMC. This second set of simulations results in a better fit to the
experimental data, suggesting that a large fraction of the observed

Gas-phase sulfuric acid has long been believed to be an important precursor
for particle formation in the atmosphere

The development of highly sensitive mass spectrometers has enabled the
detection and characterization of individual ionic clusters consisting of
only a few molecules

At the same time, modeling of particle formation has also advanced greatly in
the past few years. For the first time, simulations involving no empirical
fitting parameters give qualitatively correct predictions for the sulfuric
acid concentration dependence of cluster concentrations

Cluster formation simulations require as input the collision and evaporation
rates of clusters. The collision frequencies are usually computed simply
using classical physics, and an estimate of the evaporation rates can be
obtained by relying on equilibrium considerations and using the formation
free energies of the clusters computed by quantum chemistry. This approach
has been shown to give qualitative agreement with experiments

An alternative approach for estimating the rate constants is to start from
experimental cluster concentrations and find rate constants that reproduce
these results. This has been done previously by

In this study, measured cluster distributions are combined with detailed
cluster formation simulations, explicitly describing all possible collision
and evaporation processes. Theoretical estimates are used for the collision
rates, while all evaporation rate coefficients as well as some parameters
related to experimental details are optimized to reproduce the experimental
data. Due to the large number of unknown parameters, the fitting is done by
Monte Carlo simulation. The method is applied to measurement data from the
CLOUD experiment (

The experimental cluster distributions used in this study are from an earlier
publication from the CLOUD experiment at CERN

Cluster dynamics simulations were performed with ACDC (Atmospheric Cluster
Dynamics Code), a program that writes out the birth–death equations for a
given set of molecules and clusters and solves them by numerical integration.
Unlike in earlier implementations of ACDC where

To minimize the computational burden of solving the birth–death equations,
only negatively charged clusters were considered. Both quantum chemical
calculations and mass spectrometry measurements indicate that negatively
charged clusters with three sulfuric acid molecules or less (including the
bisulfate ion) do not take up ammonia molecules

Some of the negatively charged clusters could in principle result from
collisions of neutral clusters with negative ions, but both experimental
observations

In addition to growing by collisions with monomers or decaying by monomer
evaporations, the negative clusters can get neutralized by recombination with
positively charged ions and clusters. To keep the situation simple, the
distribution of positive clusters was not simulated explicitly, but the
overall positive ion concentration was set to match the total negative ion
concentration, and all negative ions were assumed to have the same
recombination rate coefficient of

The formation of negative ions was modeled similarly as was done by

To mimic the experimental conditions as closely as possible, each simulation
was started from a situation with non-zero sulfuric acid and ammonia monomer
concentrations and no ions. The charger ion source was switched on, and the
time evolution of the cluster concentrations was simulated, keeping the
neutral monomer concentrations constant. The experimental cluster
distributions correspond to steady-state conditions

As the measurement data consisted of steady-state concentrations, it was not
possible to fit both the collision and evaporation rates – multiplying all
rate constants by the same factor would only change the timescale of the
process but not the steady-state concentrations. Collision frequencies
between ions and polar or polarizable molecules can be approached
theoretically by considering classical electrostatic interactions. While a
closed-form analytical expression cannot be obtained even when neglecting
quantum effects, theoretical estimates for collision rates are much more
reliable than those for evaporation rates. In all the simulations presented
in this study, the collision rate constants were computed using the
parameterization of

In principle, the evaporation rates might have any values, and there is no
way to constrain even their order of magnitude based on earlier experimental
evidence or simple theoretical considerations. However, the interval in which
the evaporation rates are allowed to vary does not in practice need to be
infinitely wide. If the length of the simulation is 30 min, it does not
matter whether a cluster has a lifetime of one day or one week – it will in
any case not evaporate. On the other hand, if a cluster collides with
monomers on average once per second or once per minute, there is no effective
difference whether it has an evaporation lifetime of one millisecond or one
microsecond – it will almost certainly evaporate before it has a chance to
grow further. Even so, the range of interest for the evaporation rates spans
several orders of magnitude, and the base ten logarithms of the rates (used
as the parameters to be varied by Markov chain Monte Carlo (MCMC) instead of the rates themselves) were
sampled from the range of

The simulations also involve a large number of experiment-related parameters whose values cannot be measured directly or estimated reliably based on any fundamental theory. These were also treated as free parameters and varied using MCMC. For some of the parameters, however, at least an order-of-magnitude estimate is available, and these estimates were used for constraining the range in which the parameters were allowed to vary.

A wall loss rate of

Based on measured ion concentrations and approximate loss rates of ions, the
ion production rate due to natural ionization was estimated to be of the
order of 3 ion pairs cm

In some experiments, no ammonia was added intentionally to the chamber. While its concentration was in these cases below the detection limit of 35 ppt, some trace amount must have been present as ammonia molecules were observed in the clusters. In the simulations, two approaches were used regarding the ammonia concentration: either a constant background ammonia mixing ratio of 5 ppt was used for all these experiments, or the mixing ratio was allowed to vary separately for each of these low-ammonia experiments, and the values were sampled between 0 and 50 ppt.

It is possible that some clusters fragment inside the instrument before
detection. Weakly bound water molecules probably evaporate to a great extent

In an IMS-TOF (ion mobility spectrometer–time-of-flight mass spectrometer)
experiment, detachment of sulfuric acid molecules was observed to be
important at least for the pure trimers,

On the other hand, in another IMS-TOF experiment, larger sulfuric
acid–dimethylamine clusters

For simplicity, detachment of sulfuric acid molecules from ammonia-containing
clusters was not taken into account, although it might in reality occur to
some extent, and the removal of ammonia molecules from the clusters was
described by only four parameters: the probabilities of detecting

The effect of the above-mentioned unknown parameters (evaporation rates, ion
production rate, wall loss coefficient, background ammonia concentrations,
fragmentation probabilities) on the cluster distribution was studied by
Bayesian analysis using MCMC. (See, e.g.,

The parameters are varied using a random-walk approach, and at each step the
new parameter values (denoted as the vector

Assuming that the experimental data contain measurement errors that can be
described as uncorrelated multiplicative lognormal noise with the same
variance

Some parameters were found to have posterior distributions with more than one
local maximum. Plotting two-dimensional posterior distributions of pairs of
parameters showed in many cases L-shaped or otherwise non-convex regions of
high probability that are hard to sample using traditional methods. In order
to ensure that the random walk was able to find all the local maxima and
converged to the correct distribution, the DE-MC

Further details about the MCMC simulations are presented in the Supplement.

Schematic representation of the steps involved in the study. The green boxes show the two alternative starting points.

An overview of the simulation methods is presented in
Fig.

The starting point of the main part of the study (dark green box) were the 22 cluster distributions measured at CLOUD at varying sulfuric acid and ammonia vapor concentrations. These were used as input for an MCMC simulation, and the main output of the MCMC simulation were parameter values that reproduced most closely the measured cluster distributions. However, unlike traditional fitting procedures, MCMC gives a distribution of most-likely parameter values (called the posterior distribution) and corresponding cluster distributions instead of one best fit.

The second part of the study focused on testing the performance of the MCMC
data analysis method. First one possible set of parameter values was selected
(light green box in Fig.

Although the main result from the MCMC simulation are the distributions of
likely parameter values, it is useful first to look at the cluster
distributions corresponding to these output parameter values (referred to
later as output cluster distributions) and check how accurately the input
data are reproduced. Such comparisons are presented in
Sect.

Cluster distributions measured at CLOUD and the corresponding
modeled cluster concentrations from an MCMC simulation where only the
evaporation rates are varied and no fragmentation is allowed. “A” stands
for

The output values of the evaporation rates and fragmentation probabilities
are discussed in detail in Sect.

Even when the MCMC simulation finds a good fit to the observed distributions, the interpretation of the output parameter distributions is not always clear. The number of input data points from the CLOUD experiment is so small that unambiguous values were not reached for most of the evaporation rates. To get better insight into what conclusions can safely be drawn, Sect. S2 in the Supplement presents test simulations for synthetic input cluster distributions with known evaporation rates and fragmentation probabilities.

Figure

Using the ion production rate and wall loss constant as free parameters while still keeping a fixed background ammonia concentration does little to improve the fit. The same discrepancies remain also if the background ammonia concentrations are varied.

Figure

Cluster distributions measured at CLOUD and the corresponding
modeled cluster concentrations from an MCMC simulation where evaporation
rates, fragmentation probabilities, the ion production rate and the wall loss
rate are varied. “A” stands for

Posterior distributions of the base 10 logarithm of the evaporation
rates (in units of s

Figure

All sets of MCMC simulations give a similar result for parameters number 1, 2
and 4: the pure negatively charged sulfuric acid dimer

The distributions of some of the other evaporation rates depend strongly on
the ammonia concentration assumed for the low-ammonia experiments. For
instance, an ammonia concentration of 10 or 20 ppt (corresponding to the
case where the ammonia concentrations were treated as free parameters) would
require the

For the two cases where the background ammonia concentration is set to a
fixed value, some of the posterior distributions consist of several peaks
(see Fig.

Evaporation rates corresponding to the three MCMC simulations
presented in Fig.

The estimates extracted for the evaporation rates from the MCMC simulations
are presented in Table

For certain evaporation rates, a distinct peak is observed in the posterior distribution. Also, in this case it should be kept in mind that the true value could be anywhere within the width of the peak. As can be expected, all these well constrained evaporation rates are in the intermediate range, mostly between 1 and 100, where growth by collisions does not completely overwhelm the evaporation process, but the cluster is not so unstable that it would never collide and grow further. These clusters probably correspond to rate limiting steps on the main formation pathway.

Some of the parameters have posterior distributions with a non-zero probability density over the whole range. Some of these evaporation processes occur between clusters that are grouped together in the cluster distribution, and others are perhaps not on the main formation pathway. In any case, they do not have a strong impact on how well the modeled concentrations fit to the experimental data, and their values are therefore not constrained.

Also, evaporation rates estimated from quantum chemical Gibbs free energies

The probabilities of fragmentation processes that might occur in the inlet of the mass spectrometer were varied separately from the evaporation rates, as the process involved is different: the evaporation rates discussed in the previous section correspond to molecules evaporating spontaneously from the cluster at atmospheric pressure and a temperature of 273 K, while fragmentation in the inlet occurs when the ionic clusters are accelerated and experience high-energy collisions with neutral carrier gas molecules. In reality, the two concepts are not totally unrelated, as both processes depend on the binding energy of the cluster, but the fragmentation probability is also likely to depend on the number of vibrational degrees of freedom that can absorb energy from the collision. As the different factors determining the fragmentation probability, and even the exact conditions inside the APi-TOF inlet, remain unclear, all fragmentation probabilities were varied freely.

Posterior distributions of the fragmentation probabilities in the
mass spectrometer inlet corresponding to the experimental cluster
distributions and different options for treating the background ammonia
concentration in the experiments where it was below the detection limit and
therefore unknown. “A” stands for

Posterior distributions of the total fragmentation probabilities of
the

Figure

The posterior distribution of the dimer fragmentation probability is spread over the whole range from no fragmentation to 100 % fragmentation. While there is a peak close to 70 %, the possibility of dimers not fragmenting at all (which seems likely based on earlier experimental and theoretical evidence of the dimer being extremely stable) is not ruled out.

The trimers are found to fragment to some extent, producing both monomers and
dimers. Assuming that both dimers and trimers have very low evaporation
rates, but the tetramer is not very stable, the

Also, the pure acid tetramers and pentamers fragment, possibly even more than
the trimers, but it cannot be determined which fragmentation pathways are
most important. A large fraction of the

A Markov chain Monte Carlo (MCMC) approach is presented for determining evaporation rates from measured cluster distributions. The time evolution of the cluster population is described by birth–death equations and solved numerically. The values of the collision and evaporation rates are varied, and the obtained cluster distributions are compared to the measurements. In addition to the evaporation rates, several other poorly known parameters related to the experimental setup are varied. The method is applied to concentration distributions of negatively charged sulfuric acid–ammonia clusters measured in the CLOUD chamber in CERN.

Of the pure sulfuric acid ion clusters

The results are more ambiguous for the ammonia-containing clusters. The MCMC simulations produce several alternative sets of evaporation rates that all provide an equally good fit to the experimental cluster distributions. This inconclusiveness stems at least partly from the choice of ammonia concentrations used in the set of experiments. In more than half of the experiments, the ammonia concentrations are in an unknown narrow range below the detection limit of 35 ppt, while the other runs have ammonia concentrations in a second narrow range from 100 to 250 ppt. Repeating the MCMC simulations with a new set of experimental cluster distributions measured at ammonia concentrations distributed evenly over a wide range would most probably narrow down the estimates for many of the evaporation rates.

The observation that several alternative sets of parameter values can produce
a good fit to the same experimental data highlights the risk in using a
simplified cluster model with only one or two fitting parameters, as was done
by

Another important finding is that fragmentation in the inlet of an APi-TOF mass spectrometer may have a significant effect on the observed cluster distribution. The amount of fragmentation depends on the type of inlet that is used, and also the specific voltages and other settings that are used. However, if it is not possible to suppress fragmentation completely for some instrument type or experimental setup, it is important to at least gain some understanding of the fragmentation processes, and MCMC analysis appears to be a suitable tool for this. In this study, the mass spectrometer was assumed to have been calibrated so that there was no mass discrimination, but in the future, the mass dependent transmission efficiency of mass spectrometers could also be studied using MCMC analysis.

While definitive values could not yet be obtained for all evaporation rates, the MCMC approach is shown to be a promising new tool for analyzing cluster concentration measurements. It can give valuable information about cluster evaporation processes that cannot be observed directly. However, enough experimental data, measured over a wide range of all precursor concentrations, are needed in order to draw clear conclusions. All details related to the experimental setup must be mimicked as closely as possible in the simulations in order for the fitting parameters to have a clear physical meaning. Also, uncertainties in the measured cluster concentrations need to be taken into account in more detail in future studies. Furthermore, as cluster formation is inherently a dynamical process, the MCMC analysis would be more efficient for datasets of cluster concentrations as a function of time, instead of the steady-state distributions used here. This would also enable the fitting of collision rate constants in addition to evaporation rates.

The measured cluster distributions from Olenius et al. (2013b) were obtained from the authors of that paper.

I would like to thank CSC–IT Center for Science Ltd for computer resources, the Vilho, Yrjö and Kalle Väisälä Foundation and the European Research Council (project ERC-StG 257360-MOCAPAF) for funding, and Prof. Hanna Vehkamäki, Tinja Olenius and Heikki Haario for useful discussions and comments regarding the manuscript. Edited by: J. Abbatt Reviewed by: A. Nadykto and one anonymous referee