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 . Based on the main formation pathway in cluster formation simulations , the clusters HSO4-⋅(H2SO4)0-2, HSO4-⋅(H2SO4)3⋅(NH3)0-3 and HSO4-⋅(H2SO4)4⋅(NH3)0-4 were chosen to form the simulated system in this study. The only electrically neutral species included in the simulation were the H2SO4 and NH3 monomers.

Some of the negatively charged clusters could in principle result from collisions of neutral clusters with negative ions, but both experimental observations and quantum chemical calculations suggest that sulfuric acid–ammonia clusters are so weakly bound that their concentrations are orders of magnitude lower than the sulfuric acid monomer concentration at conditions corresponding to the experiments reported by . Therefore, the contribution of neutral clusters was not taken into account in this study. Water molecules were not modeled explicitly, but the collision and evaporation coefficients should be interpreted as effective rates averaged over the hydrate distribution of each cluster type (see for instance ).

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 1.6×10-6 cm3 s-1 with these generic positive ions. The formed neutral clusters were outside the system of interest, and their concentrations were not recorded.

The formation of negative ions was modeled similarly as was done by . Generic charger ions with the properties of O2- are first produced at a constant rate, and upon collisions with H2SO4 molecules they ionize these to form bisulfate ions. The charger ions can also be lost by recombination with positive ions. Finally, all clusters and charger ions can be lost on the chamber walls, and this was described by a size- and composition-independent wall loss coefficient.

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 , and the lengths of the individual experiments were of the order of half an hour . The modeled cluster distribution was calculated as an average of the distributions at time t1=20 min and t2=30 min after the beginning of the run. The extent to which the simulation had reached a steady state was characterized by the ratio of the concentrations at t2 and t1, calculated in each case for the cluster for which this ratio deviated most from unity. This convergence parameter was used together with the cluster concentrations to determine how well the simulations reproduced the experimental results.

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 based on classical trajectory simulations. The values for the reactions in the studied system were between 10-9 and 4×10-9 cm3 s-1.

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 -10 to 10.

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 1.7×10-3 s-1 was determined for the electrically neutral H2SO4 monomer in the CLOUD chamber . This rate decreases with increasing cluster size, but ions may have a higher loss rate. The probability of an individual cluster being lost on a wall also varies with location inside the chamber, or in practice with time as the air is continuously circulated around the chamber by large fans. As the size, charge and composition dependence of the wall losses is not known, all clusters were, for simplicity, assumed to have the same wall loss rate, and its value was sampled from the range 0 to 10-2 s-1. The size-independence of the wall loss rate may cause some uncertainty to the results, but introducing even more free parameters in order to vary the value separately for each cluster would also be problematic.

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-3 s-1 . In this study, it was sampled from the range of 0 to 10 ion pairs cm-3 s-1.

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.

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., , for an introduction to MCMC methods.) The aim of MCMC in parameter estimation is to find combinations of parameter values that reproduce the experimental data as well as possible. Instead of finding one best fit, the objective is to find a distribution of the most-likely parameter values. This is accomplished by forming a chain Z of parameter values that converges toward the desired distribution as the length of the chain increases.

An overview of the simulation methods is presented in Fig. . The same MCMC procedure (shown in orange in the figure) was used with two alternative sets of cluster distributions as input. In both cases, these cluster distributions corresponded to 22 individual experiments (or computer experiments) with varying sulfuric acid and ammonia vapor concentrations, and for each experiment the concentrations of seven cluster types were included in the distribution. In the MCMC simulation, all unknown parameters (evaporation rates etc.) were first given some random values, and these were used for running a set of 22 ACDC simulations with vapor concentrations corresponding to the input cluster distributions. The cluster concentrations obtained from the ACDC runs were compared with the input cluster concentrations, and the parameters were given new values. The new parameter values were again used to run a set of ACDC simulations, and the process was repeated over and over.

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. ). Quantum-chemistry-based theoretical predictions were used for the cluster evaporation rates, and the other parameter values were estimated based on the experiment. These parameter values (referred to later in the paper as input parameter values) were used as input for a set of 22 ACDC simulations corresponding to the same sulfuric acid and ammonia vapor concentrations as in the experimental cluster distributions. The chosen input values of the fragmentation parameters were applied to the output concentrations from these ACDC runs to get a set of 22 cluster distributions. Some random noise (see Sect. ) was added to these simulated cluster distributions to obtain synthetic “measured” cluster distributions. These, in turn, were then used as input for an MCMC simulation, and the output was again a distribution of most likely parameter values as well as corresponding cluster distributions. Since in this case the “correct answers” (that is, the input parameter values used to produce the synthetic cluster distribution) were known, the parameter distributions obtained as output from MCMC could be compared to the input values.

Figure  presents the experimental cluster distributions from CLOUD together with the output cluster distributions from an MCMC simulation where only the evaporation rates are varied and fragmentation in the mass spectrometer is not taken into account. The background ammonia concentration is set to 5 ppt, and the values reported by are used for the ion production rate and wall losses. The medians of each concentration from the output of the MCMC simulations are presented as a horizontal line, and the vertical lines span between the 2.5th and 97.5th percentiles. Comparison of the measured and simulated concentrations shows that while overall the simulated concentrations are mostly of a correct order of magnitude, the MCMC fitting does not produce the correct precursor concentration dependence for all ion cluster types. In case of the bisulfate ion HSO4- and the charged dimer HSO4-⋅H2SO4, the measured ion concentrations are notably lower in the experiments with a high ammonia concentration than in experiments with a similar acid concentration and no added ammonia, while the simulated concentrations show practically no ammonia dependence. For the larger clusters, on the other hand, the ammonia dependence is captured reasonably well. However, the sulfuric acid concentration dependence of the output cluster distributions also differs from the observed dependence for many of the larger clusters at low ammonia concentrations. This discrepancy is most prominent for HSO4-⋅(H2SO4)4⋅NH3 and HSO4-.

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  presents the output cluster distributions from an MCMC simulation where the fragmentation probabilities discussed in Sect.  are treated as free parameters. The ion production rate and wall loss constant are also varied, but all background ammonia concentrations are set to 5 ppt. Apart from a few outliers in the experimental concentrations, the agreement between the measured and modeled concentrations is remarkably good. This suggests that the poor fit in Fig.  may be explained by the concentrations observed by the mass spectrometer not corresponding directly to the ion concentrations in the CLOUD chamber, but instead to the concentrations after some of the clusters have fragmented in the inlet of the mass spectrometer. In fact, the acid and base monomer concentration dependence is very similar for the measured concentrations of the three smallest ions, HSO4-⋅(H2SO4)0-2, which would be consistent with some of the trimers being detected as monomers and dimers after having fragmented inside the instrument.

Figure shows the posterior distributions of the coefficients corresponding to logarithms of the evaporation rates. The three sets of distributions correspond to different options for treating the background ammonia concentration. Either all below-detection-limit ammonia concentrations are varied separately as MCMC parameters (green), or they are all set to 1 ppt (blue) or 5 ppt (purple). In the MCMC simulation where the background ammonia concentration is fitted, the median values for these concentrations are between 7 and 20, although the values are spread from 0 to 30 or 40.

All sets of MCMC simulations give a similar result for parameters number 1, 2 and 4: the pure negatively charged sulfuric acid dimer HSO4-⋅H2SO4, trimer HSO4-⋅(H2SO4)2 and pentamer HSO4-⋅(H2SO4)4 are stable, having evaporation rates below 1 s-1. The reason for the uniform shape of these distributions at low evaporation rates is that once the evaporation rate is much lower than the rates of any competing processes, its exact value has no effect on the cluster distribution. As discussed in the Supplement, the peak seen in some of these distributions should not be interpreted as giving a good estimate for the evaporation rate – instead, the evaporation rate can have any value below the threshold where the probability density goes to zero.

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 HSO4-⋅(H2SO4)3⋅NH3 cluster to have an ammonia evaporation rate of about 200 s-1 in order for enough pure sulfuric acid tetramers HSO4-⋅(H2SO4)3 to be observed, while the evaporation rate would need to be well below 1 s-1 if the ammonia concentration was instead 1 ppt. A similar pattern is observed for some of the other ammonia evaporation rates, and interdependencies between the different evaporation rates lead to the posterior distributions of some sulfuric acid evaporation rates also depending on how the background ammonia concentration is treated.

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. ). As described in more detail in the Supplement, the MCMC results can in fact be divided into two or three separate solutions, respectively, for the cases with background ammonia concentrations of 1 and 5 ppt. These alternative solutions correspond to different cluster types being stable and unstable, but they all still give an equally good fit to the measured cluster distributions. For instance, when assuming a background ammonia concentration of 5 ppt, the posterior distribution of parameter number 5 shows three separate peaks (purple line in Fig. ). Looking only at the sets of parameter values in the right hand side peak, it can be noted that the value of parameter number 3 always corresponds to the left hand side peak of this distribution (see Fig. S14 in the Supplement), and parameter number 8 always has a low value due to correlations between the different parameters. This set of ranges for the parameter values is denoted as solution (E), and similarly the two other peaks in the distribution of parameter number 5 correspond to solutions (C) and (D). The observation that the distributions can be divided into separate solutions in this way implies that, for instance, either an evaporation rate of 100 s-1 for ammonia from the HSO4-⋅(H2SO4)3⋅NH3 cluster and an evaporation rate of 3 s-1 of the pure sulfuric acid tetramer or an evaporation rate of 0.2 s-1 for ammonia from the HSO4-⋅(H2SO4)3⋅NH3 cluster and an evaporation rate of 60 s-1 of the pure sulfuric acid tetramer could produce a good fit to the experimental cluster distributions, but an evaporation rate of 100 s-1 for ammonia from the HSO4-⋅(H2SO4)3⋅NH3 cluster and an evaporation rate of 60 s-1 of the pure sulfuric acid tetramer would not reproduce the data. For the simulations with an ammonia concentration of 1 ppt, the separate solutions (A) and (B) correspond to the two peaks in the distribution of coefficient number 6.

The estimates extracted for the evaporation rates from the MCMC simulations are presented in Table . As discussed above, only an upper limit can be determined for some evaporation rates, and it should be noted that the actual value could equally well be just below this limit or several orders of magnitude lower. For example, the rate at which HSO4-⋅H2SO4 dimers are lost through collisions with neutral sulfuric acid molecules is between 0.04 and 1.3 s-1 in the different experiments. If the evaporation rate of the dimer is lower than this, the dimers will practically never evaporate before colliding with an H2SO4. If the evaporation process never happens, its rate cannot be expected to be determined based on the measurements. In order to constrain these low evaporation rates more tightly, experiments with very low but well quantified precursor concentrations would be needed, resulting in a lower rate for the competing growth process, but external losses and collisions with positive ions would still limit the range of evaporation rates that can be determined.

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 are presented in Table  for comparison. The theoretical evaporation rates have an uncertainty of one or two orders of magnitude, as they depend exponentially on the stepwise cluster formation energies, which have an uncertainty of 1–2 kcal mol-1. For the three smallest pure sulfuric acid clusters, HSO4-⋅(H2SO4)1-3, the quantum-chemistry-based evaporation rates are in good agreement with the values determined from analyzing the experimental data. The pure acid pentamer HSO4-⋅(H2SO4)4, on the other hand, is predicted by quantum chemistry to have an evaporation lifetime of only 5 ms, while the analysis of the experimental data suggests that it has a very low evaporation rate (and hence a very long evaporation lifetime). In case of the ammonia-containing clusters, the MCMC simulations with different options concerning the background ammonia concentration, as well as the different alternative solutions from the simulations, give different ranges of most likely values of the evaporation rates, some of which agree better and some worse with the theoretical estimates.

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.

Figure  shows posterior distributions for the studied fragmentation probabilities. For the larger pure acid clusters HSO4-⋅(H2SO4)2-4, several different fragmentation processes are considered, and their probabilities are presented separately in Fig. . The posterior distributions of the overall fragmentation probabilities (that is, the sums of the probabilities of all fragmentation processes in which a given cluster can be lost) of these clusters are shown in Fig. . For the MCMC simulations with a fixed background ammonia concentration, the distributions corresponding to the alternative solutions (see previous section) are shown in Sect. S3 of the Supplement.

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 HSO4- ions that are formed from charging H2SO4 molecules will quickly gain first one and then a second acid molecule and, as the next growth step is slower, accumulate to form a high concentration of trimers. If a notable fraction of these trimers fragment both into monomers and dimers, most of the monomers and dimers that are detected may actually be fragmentation products from trimers, as was also observed experimentally by . This would mean that the actual concentrations of negatively charged monomers and dimers in the chamber cannot be measured, preventing the accurate determination of the dimer evaporation rate and fragmentation probability. This scenario is in good agreement with the observations that the dimer fragmentation probability cannot be determined and only a relatively high upper limit can be found for the dimer evaporation rate.

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 HSO4-⋅(H2SO4)3-4⋅NH3 clusters probably lose the ammonia molecule before detection, although the exact shape of the posterior distributions depends on how the low ammonia concentrations are treated in the MCMC simulation. The results for the probability of the clusters containing two or more ammonia molecules losing all of them, on the other hand, is almost independent of the simulation options, and only a small fraction of these clusters are detected as pure acid clusters.