Stochastic Effects in H2SO4-H2O Cluster Growth

Aerosol Science and Technology

ISSN: 0278-6826 (Print) 1521-7388 (Online) Journal homepage: https://www.tandfonline.com/loi/uast20

Stochastic effects in H2SO4-H2O cluster growth

Christoph Köhn, Martin Bødker Enghoff & Henrik Svensmark

To cite this article: Christoph Köhn, Martin Bødker Enghoff & Henrik Svensmark (2020): Stochastic effects in H2SO4-H2O cluster growth, Aerosol Science and Technology, DOI: 10.1080/02786826.2020.1755012 To link to this article: https://doi.org/10.1080/02786826.2020.1755012

Accepted author version posted online: 15 Apr 2020. Published online: 29 Apr 2020.

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Stochastic effects in H2SO4-H2O cluster growth

Christoph Kohn€ , Martin Bødker Enghoff , and Henrik Svensmark National Space Institute (DTU Space), Technical University of Denmark, Kgs Lyngby, Denmark

ABSTRACT ARTICLE HISTORY The nucleation of sulfuric acid-water clusters plays a significant role in the formation of Received 8 January 2020 aerosols. Based on a recently developed particle Monte Carlo (MC) Code, we analyze how Accepted 6 April 2020 the growth of sulfuric acid-water clusters is influenced by stochastic fluctuations. We here EDITOR consider samples of H2SO4-H2O clusters at T ¼ 200 K with a relative humidity of 50%, with – – – Jim Smith particle concentrations between 105 and 107 cm 3 in volumes between 10 6 and 10 2 cm3. We present the temporal evolution of the formation rate and of the size distribution as well as growth rates and the onset time of the nucleation above a given cluster size with and without constant production of new monomers. Clear evidence is revealed by the MC code that fluctuations result in a faster growth rate of the smallest clusters compared to deter- ministic continuum models that do not contain the stochastic effects. The faster growth of small clusters in turn influences the growth of larger clusters. Depending on the volume size, the onset time for clusters larger than 0.85 nm varies between 1000 s and 20,000 s for n ¼ 105 cm–3 and between 10 s and 100 s for n ¼ 107 cm–3.

1. Introduction respectively, and C the monomer concentration. Recently it was shown that the inclusion of stochastic On a global scale approximately half of cloud condensa- fluctuations to the size distribution can significantly tion nuclei (CCN)—the seeds for cloud droplet forma- increase the calculated growth rates of sub 5- tion—originate from nucleation and condensation of nanometer sized aerosols (Olenius et al. 2018). low-volatility gases in the modern day atmosphere Stochastic fluctuations can be understood as the ran- (Merikanto et al. 2009;YuandLuo2009); this number dom collisions between molecular clusters (and corre- increases to about two thirds for the pre-industrial atmos- sponding evaporations) which happen outside of the phere (Gordon et al. 2017). Since newly formed aerosols average growth of the system given by the term bC are small in size (1 nm) and thus have a high mobility, c: These events are not included in the continuous they have a high risk of being scavenged by larger particles GDE (Equation (1)), which forms the foundation for (Pierce and Adams 2007) before they can grow to sizes many aerosol growth models (Olenius et al. 2013; relevant for cloud formation. The faster the aerosols can Trostl€ et al. 2016). grow to larger sizes the higher their probability of survival Not including the stochastic term can thus have so the dynamics of both the formation and growth of new the risk of underestimating the potential of nucleated aerosols are therefore important to understand the global aerosols to form CCN, but this is still a relatively CCN concentration (Pierce and Adams 2009). unexplored area. This work is the first application of a Traditionally the dynamics of aerosol populations recently developed 3D particle Monte Carlo model, are described by the general dynamic equation (GDE). tracking individual clusters. Such a model is capable The continuous equation only treating condensation of capturing the individual behavior of each cluster and evaporation (thus neglecting coagulation, losses to and is therefore suitable to investigate the effects of other processes, and production of new monomers) is: stochastic fluctuations on a developing aerosol popula- @cði tÞ @ , ¼ ½ðbðiÞC cðiÞÞcði tÞ tion whilst traditional models such as the classical @t @i , (1) thermodynamic nucleation theory (Hamill et al. 1982), where c is the concentration as a function of size (i), kinetic models (Pirjola and Kulmala 1998; Lovejoy, b and c the condensation and evaporation constants Curtius, and Froyd 2004;Yu2006), nucleation theory

CONTACT Christoph Kohn€ [email protected] National Space Institute (DTU Space), Technical University of Denmark, Elektrovej 328, 2800 Kgs Lyngby, Denmark. ß 2020 American Association for Aerosol Research 2 C. KÖHN ET AL.

– (Vehkam€aki et al. 2002), or Atmospheric Cluster 107 cm 3 lie in the order of 102 103 s. With these Dynamics (McGrath et al. 2012) only capture macro- time steps, the diffusive length scale varies from 42 scopic or averaged quantities and are therefore not and 134 lm. Since the mean free path of sulfuric acid suited to deal with the stochastics of particle growth. molecules in air is approximately 10–60 nm (Seinfeld The model used was previously described in Kohn,€ and Pandis 2006), hence significantly smaller than Enghoff, and Svensmark (2018) with modifications as 1 lm, the justification of the diffusion approach in noted in Section 2 where we also describe the setup this study is justified. and simulation runs. In Section 3, we present results After every time step, we check whether clusters on the temporal evolution of the size distribution and evaporate or whether two clusters collide, and if so, its dependence on simulation parameters such as determine the mass and volume of the merged cluster. monomer density and simulation volume. Modeled Throughout this article, we connect the particle i growth rates are compared with those expected from size to the number of H2SO4 molecules in the cluster the continuous GDE and we analyze the condensation (Yu 2005). In this sense, the particle radius of clusters rates in an attempt to determine the critical cluster with i monomers is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi size. Finally, we summarize and conclude our study in mðiÞ Section 4. In addition, we describe details about the Rði TÞ¼ 3 3 , p.ði TÞ (6) parallelization of our computational model, which is 4 , encoded in Python, in Appendix A. Implicitly including the size dependent amount of

H2O molecules found at a relative humidity of approx. 2. Modeling 50%. The mass and the density for temperatures between 233 and 323 K are given as in Vehkam€aki We model the motion and clustering of hydrated sul- et al. (2002): furic acid clusters as described in detail in Kohn,€ kg kg Enghoff, and Svensmark (2018) where a cluster 0:098 0:018 i i MFðiÞ mðiÞ¼ mol i þ mol kg (H2SO4)i-(H2O)j consists of H2SO4 molecules and NA NA MFðiÞ implicitly includes a size dependent amount of H2O (7) molecules. Here we recapitulate the main features of kg our simulations: We trace individual particles of given .ði, TÞ ¼½0:7681724 þ 2:184714 MFðiÞ cm3 size and update their position through pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 7:163002 MFðiÞ2 44:31447 MFðiÞ3 rðt þ DtÞ¼rðtÞþ 2DðRÞDtG (2) þ 88:75606 MFðiÞ4 with the size-dependent diffusion coefficient D(R) and 75:73729 MFðiÞ5 þ 23:43228 MFðiÞ6 the time step Dt þ T½K ð0:001808255 0:009294656 MFðiÞ 2 3 2 0:03742147 MFðiÞ þ 0:2565321 MFðiÞ Rð1Þ DðRÞ¼D 4 5 0 R (3) 0:5362872 MFðiÞ þ 0:4857736 MFðiÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:1629592 MFðiÞ6ÞþT½K 2 3 3 2 kBT 1 ð : þ : MFðiÞ D ¼ (4) 0 000003478524 0 00001335867 0 p3mð Þ 2 3 1 4PRð1Þ þ 0:00005195706 MFðiÞ2 0:0003717636 1 MFðiÞ3 þ 0:0007990811 MFðiÞ4 DtՇ pffiffiffiffiffi (5) 3 2 : MFðiÞ5 þ : 2D0 n 0 000745806 0 000258139 1 for molecules of size R where Rð1Þ¼0:329 nm and MFðiÞ6Þ 1000 mð Þ¼ : 25 1 2 0033 10 kg are the initial size and the (8) mass of single sulfuric acid molecules without any 23 –1 23 with the Avogadro constant NA 6:022 10 mol attached water molecules. kB 1:38 10 J/K is the Boltzmann constant and P ¼ 1 bar the ambient pres- and the mole fraction 0:2097 sure. For T ¼ 200 K the diffusion coefficient is D0 MFðiÞ¼0:4505 i (9) : 7 2 G 8 96 10 m /s. is the Gaussian random number which is a powerlaw fit to Fig. 2b of Yu (2005) for a G ¼ð. / h . / h . hÞ . ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficos sin , sin sin , cos with temperature of approx. 300 K for a humidity of 50%. 2 log ðr1Þ, / ¼ 2pr2, h ¼ arccosð2r3 1Þ, ri 2½0, 1Þ: Since we here investigate the nucleation of clusters at The typical time steps for densities between 105 and 200 K only, we multiply Equation (9) with a factor of AEROSOL SCIENCE AND TECHNOLOGY 3

0:8 according to the same figure in Yu (2005). As 1.2 an overview, Figure 1 illustrates Equation (6) showing 1 i the relation between the number of H2SO4 molecules and the radius R of such clusters. 0.8

Particles collide if the difference between their posi- 0.6

tions is smaller than the sum of their radii. Once two R(i) [nm] particles collide, we merge them and update their 0.4 mass as well as size according to Equations (6) 0.2 and (7). During their lifetime, clusters can also 0 evaporate and emit a monomer. After every time step, 0 5 10 15 20 25 30 we check for evaporation through r 1 exp ðcDtÞ i where r is a uniform random number r 2½0, 1Þ and Figure 1. The radius (Equation (6))ofH2SO4-H2O clusters as a where the evaporation rate (Yu 2005) is: function of the number i of H SO molecules. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4 pk Tðmð ÞþmðiÞÞ 6 2 3 8 B 1 2 1 size V ¼ 10 -10 cm such that for each particle cðiÞ¼ ðÞRð1ÞþRði, TÞ na solði, TÞ mð1ÞmðiÞ , concentration n we run simulations with N ¼ 0 2M rði, TÞ n V ¼ 10 1000 initial particles. We here use peri- exp 1 .ði, TÞRTRði, TÞ odic boundary conditions such that particles which (10) leave the simulation domain on one side enter the simulation domain on the opposite side. For each set- M ¼ – where 1 98 g mol 1 is the molar mass of H2SO4, up, we run three different simulations with different R : – 8 31 J (mol K) 1 the universal gas constant, and realizations of random numbers. All the results pre- n1 the concentration a, sol of H2SO4 vapor molecules in sented in Section 3 are then averaged over these three the equilibrium vapor above a flat source and the sur- runs such that we can exclude that our findings are face tension r for temperatures between 273 and influenced by numerical noise. 373 K are determined through (Noppel, Vehkam€aki, We run simulations with and without monomer pro- € and Kulmala 2002; Vehkamaki et al. 2002): duction. In the first case, we add one H2SO4-H2O n1 ði TÞ½3 ¼ : MFðiÞ10:891 133 N monomer whenever merging two particles reduces the a, sol , m 0 0017241 ½ A N T k R total particle number below 0. Thus, we ensure that (11) the particle number and the particle concentration stay constant. However, in cases of evaporation, the emitted rðiÞ½Nm 1 ¼ 0:11864 0:11651 MFðiÞ molecule is added to the simulation domain and the 2 3 þ 0:76852 MFðiÞ 2:40909 MFðiÞ particle density slightly increases. Yet in all considered þ 2:95434 MFðiÞ4 cases, we have observed that the growth of the concentration of all particles is negligible and does not change 1:25852 MFðiÞ5 þ T½k our conclusions. In the second case, we do not add par- ð : þ : MFðiÞ 0 00015709 0 00040105 ticles to the simulation domain if the particle number : MFðiÞ2 N 0 0023995 decreases below 0; hence, the gas depletes and, if the þ 0:007611235 MFðiÞ3 0:00937386 simulation ran sufficiently long enough, we would end up with one cluster containing all the initial molecules. MFðiÞ4 þ 0:00389722 MFðiÞ5Þ All simulations are performed in Python. Since (12) such simulations can take from days up to several Because of the lack of physical data, e.g., for the equi- weeks, depending on the physical time and on the librium vapor pressure, we still use Equations (8) and number of particles, the code has been parallelized as (12) for T ¼ 200 K and multiply Equation (10) with a described in detail in Appendix A. factor of 105 for 200 K based on Fig. 4 of Yu (2005) where evaporation rates for different temperatures and 3. Results humidities are depicted. For more details, we refer to € 3.1. Dependence of the formation rate on Kohn, Enghoff, and Svensmark (2018). the volume We have performed simulations for initial concen- – – trations of n ¼ 105 cm 3 up to n ¼ 107 cm 3 particles Figure 2 shows the formation rate of clusters with a with one central sulfuric acid molecule in domains of radius larger than 0.85 nm at 200 K for different initial 4 C. KÖHN ET AL.

3 20 10 V=10-4 cm3 -3 3 -3 3 V=10 cm V=10 cm 2 -2 3 V=10-2 cm3 10 V=10 cm 15 Param. (B1) ]

] 1

-1 10 -1 s s -3 -3 10 100 [cm [cm

0.85 -1 0.85

J 10 J 5 10-2

0 10-3 0 50000 100000 150000 200000 0 50000 100000 150000 200000 t [s] t [s]

350 3 V=10-5 cm3 V=10-4 cm3 V=10-4 cm3 300 -3 3 -3 3 V=10 cm 2.5 V=10 cm Param. (B1) 250 ] ] 2 -1 -1 s s

200 -3 -3 1.5 [cm [cm 150 0.85 0.85 J

J 1 100

50 0.5

0 0 0 20000 40000 60000 0 25000 50000 t [s] t [s]

90000 2000 V=10-6 cm3 -5 3 -5 3 V=10 cm 80000 V=10 cm -4 3 V=10-4 cm3 V=10 cm 70000 Param. (B1) 1500 ] 60000 ] -1 -1 s s -3 50000 -3 1000 [cm 40000 [cm 0.85 0.85 J 30000 J 20000 500 10000 0 0 0 200 400 600 800 1000 1200 1400 1600 100 101 102 103 104 t [s] t [s]

Figure 2. The formation rate of H2SO4-H2O clusters larger than R ¼ 0.85 nm for different initial densities with (first column) and without (second column) constant particle production. The dotted lines show the formation rates (20) from the parameterization summarized in Appendix B.

cluster concentrations and for different domain vol- formation rate when the total particle number umes. The size is chosen since R ¼ 0.85 nm is compar- decreases in time as a result of particle merging. able to the size of nucleated clusters for the sulfuric Comparing the panels within each column shows that acid-water system used in several large parameteriza- increasing the initial particle concentrations increases tions (Dunne et al. 2016; Gordon et al. 2017; Tomicic, the formation rate. Additionally, the particles start to Enghoff, and Svensmark 2018), although measure- cluster earlier for increased particle concentrations. In ments at even smaller sizes also are possible the left column, hence for simulations with constant (Kontkanen et al. 2017). The left column of Figure 2 particle production, the dotted lines show the parame- shows the formation rates when keeping the particle terized formation rate deducted from experiments by density constant whilst the second column shows the Dunne et al. (2016), see Appendix B. As shown in AEROSOL SCIENCE AND TECHNOLOGY 5

Kohn,€ Enghoff, and Svensmark (2018), our simulation However, we also see fluctuations amongst the three results develop asymptotically against the parameter- different simulations for each combination (n, V) ized formation rate and agree well within one order based on the stochastic nature revealed by the Monte – of magnitude. Carlo approach. For V ¼ 104 cm3 and n ¼ 106 cm 3 Each panel shows the formation rates for one par- with particle production, the onset time in two simu- ticle concentration, but for different domain sizes. lations is approximately 1300s whilst it is 600 s for – – Note that the formation rate for n ¼ 106 cm 3 and the third one. For n ¼ 105 cm 3 and the same volume, V ¼ 103 cm3 is initially so small such that it is the onset times in three different simulations amount barely visible in panels c and d. The comparison for to approx. 1100, 900, and 400 s. Hence, although in – different volumes demonstrates that the formation two simulations the onset time for n ¼ 106 cm 3 is – rates tend to one asymptotic value for each particle larger than for n ¼ 105 cm 3, we have discovered one concentration irrespective of the domain size. simulation where clusters above 0.85 nm occur faster n ¼ 5 –3 However, the temporal evolution of J0:85 depends sig- than in two simulations for 10 cm . Since we nificantly on V. Clusters above 0.85 nm appear first in are limited by the runtime of our Monte Carlo code, small domain sizes, then later for larger domain sizes. we cannot run simulations with larger volumes. Additionally in small domains, the formation rates However, Figure 2 shows that for the largest volumes, increases suddenly, peaks and then decreases toward the formation rates grow more smoothly than for the the asymptotic value whereas the rate in larger vol- smaller volumes. Therefore, we believe that the onset umes grows rather monotonically toward the asymp- times are sufficiently converged for the larg- totic formation rate. est volumes. n ¼ 5 –3 Figure 3 shows the onset time tO for clusters above For 10 cm , Table 1 summarizes the onset 0.85 nm, i.e., the time when the first cluster larger times for clusters of several different sized (i) in dif- than 0.85 nm appears, for different volumes. Each dot ferent domain volumes with and without particle pro- shows the onset time for one particular simulation duction as an average of the three runs for each (three simulations were made for each setting). Note setting. For small clusters, i 2 and i 3, the onset that 12 molecules are needed to form clusters above time is comparable for all volumes (irrespective of the i 0.85 nm; thus in those simulations without particle particle production). However, for 5 and more sig- i production where the initial particle number is below nificantly for 7, the onset time increases for larger 12, clusters above 0.85 nm cannot appear. The general volumes which proceeds until clusters larger than i : tendency shows that the onset time is smallest for 0.85 nm or equivalently 12 In the following, we discuss what causes these dif- small domain sizes and large for larger domains. – ferences by the way of the example for n ¼ 105 cm 3: Figure 4 shows the size distributions after different 5 10 time steps with and without particle production. It

104 demonstrates that cluster growth is more efficient in small domains. After 5 103 s, the largest cluster has a 103 size of i ¼ 27 in a domain of V ¼ 104 cm3 when par- [s]

o ticles are added to the simulation domain. Note that t 2 10 n=105 cm-3 n ¼ 5 –3 n=106 cm-3 for a particle concentration of 10 cm and a 7 -3 1 n=10 cm V ¼ 4 3 10 n=105 cm-3, no production volume of 10 cm , there are only ten particles n=106 cm-3, no production n=107 cm-3, no production in the simulation domain. However, we add one 100 10-6 10-5 10-4 10-3 10-2 monomer after every collision; hence clusters can V [cm3] become larger than i ¼ 10. After 5 103 s, the Figure 3. The onset time of clusters larger than 0.85 nm for second largest clusters are found in a domain of different concentrations and volumes. size V ¼ 103 cm3, followed by the domain of size

Table 1. The onset time [s] of clusters larger than size i for n ¼ 105 cm–3 and T ¼ 200 K in different volumes. V ¼ 104 cm3 V ¼ 103 cm3 V ¼ 102 cm3 V ¼ 103 cm3,no V ¼ 102 cm3,no i 2 1.19 ± 0.78 2.57 ± 1.28 0.99 ± 0.03 1.67 ± 0.95 1.33 ± 0.47 i 3 10.39 ± 6.13 7.71 ± 3.23 18.32 ± 15.70 9.01 ± 5.89 11.67 ± 9.46 i 5 61.10 ± 16.35 177.88 ± 43.10 518.135 ± 98.115 210.01 ± 65.16 223.01 ± 105.76 i 7 132.97 ± 61.90 471.25 ± 108.60 2038.91 ± 318.89 954.01 ± 578.15 1895.01 ± 796.19 The first three columns show results for simulations with monomer production, the last two columns show results without monomer production. 6 C. KÖHN ET AL.

3 10 103 -4 3 V=10 cm V=10-4 cm3 -3 3 V=10 cm V=10-3 cm3 2 -2 3 -2 3 10 V=10 cm 102 V=10 cm -3 3 V=10 cm , no source term V=10-3 cm3, no source term -2 3 V=10 cm , no source term V=10-2 cm3, no source term 1 10 101 N(i) N(i)

0 10 100 // //

0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 i i

3 3 10 -4 3 10 V=10 cm -4 3 -3 3 V=10 cm V=10 cm -3 3 V=10-2 cm3 V=10 cm 2 V=10-3 cm3, no source term 2 -2 3 10 -2 3 10 V=10 cm V=10 cm , no source term -3 3 V=10-2 cm3, mono V=10 cm , no source term -2 3 V=10 cm , no source term 101 101 N(i) N(i)

100 100 // //

0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 i i

Figure 4. The size distribution of all clusters for n ¼ 105 cm–3 and different volumes after different time steps. Panel c has a simulation without coagulation (labeled ‘mono’).

V ¼ 102 cm3. In large domain, the size distributions involving at least one monomer were allowed. It is are much smoother than in small domains because of seen that for the initial steps coagulation is not the larger number of particles. The same tendency important for this number concentration, whereas the holds if no particles are added to the simulation simulation with similar volume including coagulation domain. Larger clusters are found in smaller domains reaches larger cluster sizes. whereas the size distribution is smoother in larger Figure 5 shows the size distributions for n ¼ 106 – – domains. In comparison to the simulations where par- cm 3 after 5000s (a) and for n ¼ 107 cm 3 after 200 s ticles are added, the clusters are smaller because of (b). We observe the same tendencies which we have – the limited number of particles. In this case, the size observed for n ¼ 105 cm 3: Small domain sizes favor of the largest cluster can never be larger than the ini- larger clusters whereas large volumes imply rather tial number of particles. smooth size distributions. In panel b, a single simula- The same tendency holds when the simulation pro- tion without coagulation (similarly to the case in gresses in time (panels b–d). In all considered cases, Figure 4c) has been added. Here the effect of lack of the largest clusters are found in small simulation vol- coagulation is seen more clearly on the size umes. The size distributions in larger domains keep distribution. their smooth shape as the cluster size grows. In panel To understand this pattern, Figure 6 shows the c, a single simulation (the other results are averaged number of clusters of sizes i ¼ 2, 5, 7, and 12 normal- N over three simulations) has been added where coagu- ized to the initial particle number 0 as a function of – lation was disabled, meaning that only collisions time for n ¼ 105 cm 3 for different volumes with and AEROSOL SCIENCE AND TECHNOLOGY 7

3 3 10 10 -6 3 -5 3 V=10 cm V=10 cm -5 3 -4 3 V=10 cm V=10 cm V=10-4 cm3 2 -3 3 2 V=10-5 cm3, no source term 10 V=10 cm 10 -4 3 -4 3 V=10 cm , no source term V=10 cm , no source term V=10-4 cm3, mono V=10-3 cm3, no source term 101 101 N(i) N(i)

100 100 // //

0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 i i

Figure 5. The size distribution of all clusters for (a) n ¼ 106 cm–3 and (b) n ¼ 107 cm–3 and different volumes. Panel b has a simulation without coagulation (labeled ‘mono’).

0.2 0.2

0.15 0.15 0 0

0.1 0.1 /N /N

0.05 0.05

0 0 0 50000 100000 150000 200000 0 50000 100000 150000 200000 t [s] t [s]

0.15 0.15 V=10-4 cm3 V=10-3 cm3 V=10-2 cm3 V=10-3 cm3, no source term V=10-2 cm3, no source term

0 0.1 0 0.1

/N 0.05 /N 0.05

0 0 0 50000 100000 150000 200000 0 50000 100000 150000 200000 t [s] t [s]

5 –3 Figure 6. The number of clusters of size i normalized to the initial particle number N0 as a function of time for n ¼ 10 cm . without particle production. Panel a shows that the keep the total particle number constant or whether we normalized number of dimers fluctuates more signifi- take gas depletion into account. For larger cluster sizes, cantly in small domains irrespective of whether we the fluctuations decrease for V ¼ 103 102 cm3. 8 C. KÖHN ET AL.

Table 2. The mean particle number hNðiÞi (13) for different sizes i. V¼104 cm3 V¼103 cm3 V¼102 cm3 103 cm3, w/o 102 cm3, w/o i ¼ 2 1.07 ± 2.25 2.86 ± 1.80 53.48 ± 20.14 2.03 ± 3.64 22.85 ± 4.76 i ¼ 5 1.02 ± 2.55 2.72 ± 2.28 64.78 ± 46.68 1.87 ± 4.21 9.38 ± 2.79 i ¼ 7 1.05 ± 3.42 1.84 ± 1.87 20.86 ± 19.09 1.69 ± 3.51 1.78 ± 1.35 i ¼ 12 1.11 ± 3.08 1.66 ± 1.66 12.98 ± 14.55 1.09 ± 2.04 1.29 ± 1.53 Columns four and five show hNðiÞi without adding particles.

3 perspective it is the maximum in the Gibbs free 10 γ energy DG of formation vs r plot occurs (Curtius 2 β 10 βw/ coagulation, 1 w/ coagulation, 2 2006) or the saddlepoint in a two-component system. 1 β 10 βw/ coagulation, 3 From a kinetic standpoint the critical cluster size is w/o coagulation 0 where the collision rate equals the evaporation rate.

] 10 -1 -1 Note that these two sizes are not always exactly the [s 10 γ

, same (Nishioka 1995). This size is extremely depend- β -2 10 ent on temperature due to the evaporation rate and 10-3 on the concentration of the condensing gas, due to

10-4 the collision rate. One way to find the critical cluster size for a given set of parameters is to compare colli- 10-5 5 10 15 20 25 30 sion and evaporation rates (Yu 2005). In our model i the evaporation rates are hard coded, but collision Figure 7. The evaporation rate (10) and the production rate rates are an emergent property so it would be interest- (16) as a function of cluster size i with and without coagula- ing to try to determine the critical cluster size based 7 –3 4 3 tion for n ¼ 10 cm and V ¼ 10 cm . Taking coagulation on model output. into account, we present results from three simulations with After a simulation has reached equilibrium, i.e., different random numbers. dNðiÞ=dt 0 for all relevant sizes i, this equilibrium translates to Only for V ¼ 104 are fluctuations still very large 0 ¼ cðÞi þ 1 nði þ 1ÞþhðÞi 1 nði 1ÞhðiÞnðiÞ compared to larger volumes. Table 2 shows the mean particle number cðiÞnðiÞ X hNðiÞi ¼ 1 Nði tÞ (15) T , (13) t where c is the evaporation rate (Equation (10)), h the b nð Þ nðiÞ¼NðiÞ=V averaged over time of the whole simulation run with condensation rate ( 1 ) and the i the standard deviation concentration of clusters of size . Approximating sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hði 1ÞhðiÞ and cðiÞcði þ 1Þ allows us to solve X h DNðiÞ¼ 1 ðÞNði tÞhNðiÞi 2: for the production rate T , (14) t NðiÞNði þ 1Þ hðiÞ¼cðiÞ (16) Nði 1ÞNðiÞ Table 2 illustrates that for V ¼ 10 4 cm3 the standard deviation exceeds the mean particle number. For which depends on the evaporation rates and on the larger volumes, the standard deviation becomes com- cluster concentrations and allows us to approximate h parable or smaller than the mean particle number. from the output of the simulations. Note that this Such fluctuations in small simulation domains imply analysis is related to the standard Becker-Doring€ der- € that in small simulation domains there is a higher ivation (Becker and Doring 1935; Wattis 1999). probability to create first dimers and subsequently Figure 7 shows the evaporation rate (Equation (10)) and the condensation rate (Equation (16)) with larger clusters; hence in domains with small volumes, – and without coagulation for n ¼ 107 cm 3 and V ¼ the production of large clusters is accelerated. – 104 cm 3 for which the formation rate for clusters larger than 0.85 nm is depicted in Figure 2e. It shows 3.2. Critical cluster size that in all considered cases the condensation rate The critical cluster size is a central concept in nucle- oscillates around the evaporation rate. Note that we ation as it is the size where the cluster becomes stable still have stochastic fluctuations in the output of our and is said to nucleate. From the thermodynamic model, therefore ðNðiÞNði þ 1ÞÞðNði 1Þ NðiÞÞ 1 AEROSOL SCIENCE AND TECHNOLOGY 9

h 2 also fluctuates explaining the oscillation of ; in mod- 10 -4 3 N i V=10 cm els, where the ( ) are calculated deterministically, V=10-3 cm3 -2 3 fluctuations would tend to zero. Note also that in 101 V=10 cm V=10-2 cm3, mono some cases Nði þ 1Þ > NðiÞ or equivalently NðiÞ > -3 3 ] V=10 cm , no source term -2 3 Nði 1Þ such that h < 0 and hence h is not visible on -1 100 V=10 cm , no source term GRcond the logarithmic y-axis. Apart from these constraints, we are able to determine the (first) intersection point 10-1 GR [nm h between the condensation and evaporation rate indi- -2 cating the critical cluster size. For the simulation with- 10 out coagulation, the critical cluster size lies at approx. -3 i ¼ 10 2 (keeping the stochastic effects in mind) whilst it 0 0.5 1 1.5 2 i ¼ i ¼ i ¼ i lies at 1 2, 2 2, and 3 7 averaging to 4 rp [nm] which is qualitatively in good agreement with the data Figure 8. The comparison of the growth rate GRcond (17) omit- presented in Yu (2005). In theory, this method would ting stochastic effects with the apparent growth rates (18) for allow us to determine the critical cluster size for all of different volumes with and without particle production includ- – the presented simulations if we were not limited by ing a single simulation without coagulation for n ¼ 105 cm 3. the long runtime of the simulations. Naturally, coagulation plays an important role for the critical cluster Equation (17) is calculated analytically and therefore size (Ehrhart and Curtius 2013; Malila et al. 2015). only depends on the particle concentration n, but not Yet, with all the stochastic effects, we are not able to on the particle number, i.e., on volumes and source clearly reveal its effect. Note that fluctuations in tem- terms. Here . is the density (Equation (8)), c the perature would lead to additional fluctuations in the evaporation frequency (Equation (10)), and for b we critical cluster size (McGraw and LaViolette 1995). use the analytic condensation rate (Yu 2005): However, since the temperature is kept constant in sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi our simulations, this effect is negligible for our 8pkBTðmð1ÞþmðRðiÞÞÞ bðRðiÞÞ ¼ ðÞRð1ÞþRðiÞ 2: considerations. mð1ÞmðRðiÞÞ If the stochastic processes contribute significantly (19) to the formation rate of small clusters they could reduce the critical cluster size from a kinetic view- Figure 8 shows the presence of two different point. Since the critical cluster size is defined from regimes. For clusters smaller than approximately GR (DG) this phenomenon is not a true reduction of the 1.2 nm, cond tends to zero suggesting that clusters GR critical cluster size, but could perhaps be considered would not grow at all. Above 1.2 nm, cond grows as an effective critical cluster size. smoothly toward an asymptotic value of approxi- 2 –1 mately 10 nm h . The growth rate GRapp (Equation GR 3.3. Growth rates (18)) appears to behave similarly to cond for clusters larger than 1.2 nm where it tends to a value close –3 – For n ¼ 105 cm , Figure 8 compares the growth rate to 102 nm h 1. GR GR ðRÞ¼ 1 ðÞbðRðiÞÞn cðRðiÞÞ mðRðiÞÞ However, for clusters smaller than 1.2 nm, cond cond GR GR 2p.ðRðiÞÞR2 and app differ significantly. Whilst cond tends to GR 2 –1 2 zero, app varies between 10 nm h and 10 nm (17) – h 1 which is due to stochastic effects neglected in omitting stochastic effects with the apparent growth GR cond. Therefore, stochastic effects give a natural rate explanation why particles initially grow and form 2DR larger clusters. If stochastic effects were omitted, the GRappðRÞ¼ (18) Dt growth rate becomes negative suggesting no cluster which is the change of size per time interval and growth at all. In almost all considered cases, except therefore gives us the opportunity to determine the for some deviations for V ¼ 10 2 cm3, the growth speed of cluster growth in contrast to the formation rates are highest for approx. 0.3 nm, the size of clus- rate of clusters of a given size or the size distribution ters with one central sulfuric acid molecule, and for a fixed time step. Whereas Equation (18) is calcu- decreases as a function of time. Consistent with the lated from our simulation results and therefore distin- fluctuations and size distributions discussed in guishes different volumes and particle production, Section 3.1, the growth rates are largest for small 10 C. KÖHN ET AL.

– volumes, i.e., a small number of initial particles, and of n ¼ 107 cm 3 in a volume of 104 cm3,we decreases with increasing domain size which is an have here determined this critical cluster size to be artifact of the stochastic fluctuations for small vol- approximately i ¼ 6 which is in agreement with the umes. Finally a single simulation (the other results critical cluster size determined in Yu (2005). In the- shown are averaged over three simulations) without ory, we could potentially calculate the critical cluster coagulation (meaning that only collisions involving a sizes for all considered cases. However, we are here monomer are allowed) using the largest volume limited by the long runtime of our calculations and (V ¼ 102 cm3) has been performed showing that in by the computational infrastructure such that the con- this regime it is not coagulation that causes the centrations have not reached equilibrium yet. observed effect. However, in future, computers with faster CPUs, a translation of the applied Monte Carlo code to GPUs or using super computers with a multitude of CPUs 4. Discussion and conclusion would allow us to run simulations faster and therefore We have simulated the motion and growth of sulfuric reach equilibrium faster. acid-water clusters at 200 K and 1 bar for densities of In conclusion, we have observed that growth rates – n ¼ 105–107 cm 3 in volumes between 106 cm3 and and formation rates are significantly larger in small 102 cm3. We have observed that growth rates for volumes where stochastic effects are dominant but smaller clusters are significantly larger when stochastic that these effects exist across all investigated domain effects are included. Thus, the occurrence onset time sizes. Stochastic effects are not visible at all in models of clusters above 0.85 nm varies between approx. which are based on averaged quantities only and the – 1000 s and 20,000 s for n ¼ 105 cm 3 and between 10 inclusion of stochastics would more accurately portray – and 100 s for n ¼ 107 cm 3. Since the formation rates atmospheric processes enhancing the very early stages decrease as temperatures increase due to faster evap- of aerosol formation and could thus affect the forma- oration rates, we conclude that the times for nucle- tion of even larger particles. ation calculated in Section 3.1 can be seen as lower This study was the first application of a recent 3D € limits only for temperatures relevant to the Monte Carlo code (Kohn, Enghoff, and Svensmark lower atmosphere. 2018). In future work, we will extend this code with We have seen that the temporal evolution of the further species, including ions, and optimize the run- cluster growth depends not only on the initial particle time to further study the spatio-temporal dependence – concentration, but also on the volume of the domain and the influence of dipole dipole interactions on size or equivalently on the initial particle number. nucleation processes. Smaller domain sizes or equivalently a smaller number of initial particles fluctuate more significantly which Auhor contributions in turn favors a faster growth initially of small clusters All simulations were performed by C.K. Evaluations, and subsequently of larger clusters. A comparison of interpretations and writing were performed together. the growth rate of our simulations with the analytic growth rate omitting stochastic effects shows the natural demand for stochastic fluctuations to initiate the Disclosure statement growth of the very smallest clusters. There are no competing interests. The enhanced initial growth rate can be critical not only for pushing particles past the nucleation barrier, but for growing clusters to larger sizes in general. As Funding scavenging by larger particles is most efficient for the This project has received funding from the European smallest clusters this can improve the survivability of Unions Horizon 2020 research and innovation program clusters and thus improve their chance of growing to under the Marie Sklodowska-Curie grant agreement 722337. CCN sizes. In alignment with the dependence of the temporal evolution of the cluster sizes, we have presented a way ORCID h to calculate the condensation rate based on the out- € h Christoph Kohn http://orcid.org/0000-0002-7101-5889 come of the Monte Carlo particle code. By equating Martin Bødker Enghoff http://orcid.org/0000-0001- with the evaporation rate c, this allows us to estimate 8452-698X the critical cluster size. For a particle concentration Henrik Svensmark http://orcid.org/0000-0003-2843-3417 AEROSOL SCIENCE AND TECHNOLOGY 11

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Figure A1. (a) The formation rate J0:85 of clusters larger than 0.85 nm as a function of physical time for a particle concentration of n ¼ 107 cm–3 in a volume of V ¼ 104 cm3. The dashed line shows the formation rate from a simulation with one single processor, the dotted line from a parallelized simulation with 20 processors. (b) The runtime T as a function of physical time for simulations with 1 and 20 processors. The right y-axis shows the ratio of the runtime for the two different simulations. package “Multiprocessing” with the subpackage Appendix B: Parameterizations of formation “pool”. We initiate a pool of p processors where we dis- rates tribute all N particles such that each processor advances 1 J approximately the same number N p of particles. We As parameterization, we use the formation rate 0:85 of have tested the parallelization with a simulation of 1000 H2SO4-H2O clusters derived from experiments by Dunne – particles in a volume of 104 cm3, thus n ¼ 107 cm 3 run- et al. (2016) for temperatures between 208 and 292 K: ning on one processor and on twenty processors. ÀÁ p Forthisparticularset-up,Figure A1a shows the for- J½cm 3s 1 ¼ exp u v T½K =1000 n½106cm 3 (B1) mation rate of clusters above 0.85 nm of the simulation as a function of physical time. It demonstrates that we with get comparable results with respect to the formation rate implying that implementing this parallelization into u ¼ 46:3 (B2) Python does not falsify our results. Panel b shows that the runtime is approximately 16 times faster. The v ¼ 245:0 (B3) decrease in runtime does not amount to 20 because we have only parallelized the part of the code with respect p ¼ 3:62 (B4) to particle motion, collision and evaporation. In contrast, overhead procedures such as the input-output of data run for given temperature T and density n (here in units of 6 –3 only on one processor slowing down the parallelized 10 cm ) of initial H2SO4-H2O clusters with one central code slightly. sulfuric acid molecule.