A predictive model for salt nanoparticle formation using heterodimer stability calculations

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ A predictive model for salt nanoparticle formation using heterodimer stability calculations © 2021 the Authors Published version Chee, Sabrina; Barsanti, Kelley; Smith, James N.; Myllys, Nanna Chee, S., Barsanti, K., Smith, J. N., & Myllys, N. (2021). A predictive model for salt nanoparticle formation using heterodimer stability calculations. Atmospheric Chemistry and Physics, 21(15), 11637-11654. https://doi.org/10.5194/acp-21-11637-2021 2021 Atmos. Chem. Phys., 21, 11637–11654, 2021 https://doi.org/10.5194/acp-21-11637-2021 © Author(s) 2021. This work is distributed under the Creative Commons Attribution 4.0 License. A predictive model for salt nanoparticle formation using heterodimer stability calculations Sabrina Chee1, Kelley Barsanti2, James N. Smith1, and Nanna Myllys1,3 1Department of Chemistry, University of California, Irvine, CA, USA 2Department of Chemical & Environmental Engineering, University of California, Riverside, CA, USA 3Department of Chemistry, University of Jyväskylä, Jyväskylä, Finland Correspondence: Nanna Myllys ([email protected]) and James N. Smith ([email protected]) Received: 29 January 2021 – Discussion started: 22 March 2021 Revised: 29 June 2021 – Accepted: 30 June 2021 – Published: 5 August 2021 Abstract. Acid–base clusters and stable salt formation are critical drivers of new particle formation events in the atmosphere. In this study, we explore salt heterodimer (a cluster of one acid and one base) stability as a function of gas-phase acidity, aqueous-phase acidity, heterodimer proton transference, vapor pressure, dipole moment and polarizability for salts comprised of sulfuric acid, methanesulfonic acid and nitric acid with nine bases. The best predictor of heterodimer stability was found to be gas-phase acidity. We then analyzed the relationship between heterodimer stability and J4×4, the theoretically predicted formation rate of a four-acid, fourbase cluster, for sulfuric acid salts over a range of monomer concentrations from 105to 109molec cm−3and temperatures from 248 to 348 K and found that heterodimer stability forms a lognormal relationship with J4×4. However, temperature and concentration effects made it difficult to form a predictive expression of J4×4. In order to reduce those effects, heterodimer concentration was calculated from heterodimer stability and yielded an expression for predicting J4×4for any salt, given approximately equal acid and base monomer concentrations and knowledge of monomer concentration and temperature. This parameterization was tested for the sulfuric acid–ammonia system by comparing the predicted values to experimental data and was found to be accurate within 2 orders of magnitude. We show that one can create a simple parameterization that incorporates the dependence on temperature and monomer concentration on J4×4 by defining a new term that we call the normalized heterodimer concentration, 8. A plot of J4×4vs. 8collapses to a single monotonic curve for weak sulfate salts (difference in gas-phase acidity >95 kcal mol−1) and can be used to accurately estimate J4×4within 2 orders of magnitude in atmospheric models. 1 Introduction Atmospheric aerosol particles represent the largest uncertainty in our understanding of global climate through their participation in cloud formation and the absorption and scattering of radiation (Kerminen et al., 2005; Kuang et al., 2009; Lohmann and Feichter, 2005; Merikanto et al., 2009; Spracklen et al., 2008). In particular, particle formation by nucleation is still not well understood and is difficult to represent in models (Kerminen et al., 2018). One of the dominant nucleation pathways is through salt formation, where the formation of a cluster is stabilized by the interactions between acid and base molecules, which enhances particle formation (Ball et al., 1999; Kirkby et al., 2011; Kürten et al., 2016; Nadykto and Yu, 2007; Nadykto et al., 2015; Wang et al., 2018). This nucleation pathway is particularly dominant in urban environments, where anthropogenic sources for acidic and basic gases are abundant (Ge et al., 2011; Kirkby et al., 2011; Qiu and Zhang, 2013; Weber et al., 1996; Wang et al., 2020). Although sulfuric acid (H2SO4, SA) is most commonly associated with atmospheric nucleation (Ball et al., 1999; Bzdek et al., 2012; Kirkby et al., 2011; Angelino et al., 2001; Weber et al., 1995), nitric acid (HNO3, NA) and methanesulfonic acid (CH3SO3H, MSA) have been also observed to be participants and may also play important roles in the initial stages of cluster growth (Afpel et al., 1979; Barsanti et al., 2009; Mäkelä et al., 2001; Smith Published by Copernicus Publications on behalf of the European Geosciences Union. 11638 S. Chee et al.: A predictive model for salt nanoparticle formation et al., 2004, 2008; Weber et al., 1995), the latter of which we shall refer to henceforth as new particle formation (NPF). Ammonia is the most abundant base in the atmosphere and its reaction with sulfuric acid has been well studied (Bzdek et al., 2010; Glasoe et al., 2015; Weber et al., 1996). Alkylamines have also garnered attention due to their high basicity and demonstrated ability to enhance NPF more than ammonia, despite their lower atmospheric abundance (Kurtén et al., 2008; Smith et al., 2010; Temelso et al., 2018; Waller et al., 2019; Kreinbihl et al., 2020). Recently, computational efforts have focused on accurately representing the formation and growth of acid–base clusters (Smith et al., 2021). Myllys et al. (2016a) investigated the accuracy of the domain local pair natural orbital coupled cluster (DLPNO–CCSD(T)) method and found that it allows for the modeling of up to 10 molecules in a cluster, which had not been previously feasible with other highly accurate methods. The DLPNO–CCSD(T)/augcc-pVTZ//ωB97X-D/6-31++G** level of theory has become increasingly popular for modeling atmospheric processes such as cluster formation of sulfuric acid with ammonia, methylamine, dimethylamine, trimethylamine, guanidine, monoethanolamine, trimethylamine N-oxide and a variety of diamines (Myllys et al., 2016a; Ma et al., 2019; Xie et al., 2017; Myllys et al., 2020, 2018; Elm et al., 2016, 2017). This large variety in systems studied has yielded insights into the factors that determine cluster formation and growth. Generally, the enhancing efficiency of the base on heterodimer stability and NPF is known to correlate with base strength, which has been attributed to a more favorable proton transfer and the formation of essentially nonvolatile ionic salts and has been shown to be generally true for the most abundant bases in the atmosphere: ammonia, methylamine, dimethylamine and trimethylamine (Almeida et al., 2013; Elm, 2017; Myllys et al., 2019b; Barsanti et al., 2009; Shen et al., 2020; Han et al., 2020; Kürten et al., 2014; Jen et al., 2014a). For many studies that observe both cluster and nanoparticle formation and growth, pKahas been often used as the metric for basicity. However, since pKais, by definition, an aqueous measure of acidity, applying it to cluster and nanoparticle-sized systems does not take into account the drastically different environment. Indeed, in the study by Xie et al. (2017), monoethanolamine (pKa=9.5) enhanced NPF more than methylamine (pKa=10.6), despite methylamine being the stronger base according to their pKavalues (Haynes, 2014). In that study, the lack of a base strength trend was attributed to the additional hydrogen bonding sites provided by the -OH group on monoethanolamine. In addition, we have recently studied the modeled formation rates of sulfuric acid and trimethylamine-N-oxide (TMAO), guanidine or dimethylamine, where TMAO, despite its lower basicity (pKa=4.7) to both guanidine (pKa=13.6) and dimethylamine (pKa=10.7), had similar formation rates to guanidine, which were much higher than those of dimethylamine (Myllys et al., 2020; Haynes, 2014). In these studies, pKawas insufficient to predict NPF enhancement. In this study, we aim to use these computational methods to identify what molecular properties predict heterodimer stability, or more specifically the Gibbs free energy of formation of the heterodimer (1Gheterodimer), and, in turn, formulate a model to predict NPF rate. We specifically investigate the use of pKain comparison to gas-phase acidity measures to predict proton transfer in the heterodimer as well as heterodimer stability. In addition, we examine if base vapor pressure has any correlation with heterodimer stability, as sulfuric acid is often cited to participate in NPF because of its low volatility and condensation onto clusters (Weber et al., 1996; Ball et al., 1999; Sipilä et al., 2010). Finally, we also calculate the dipole moment and polarizability of the studied base molecules to see if, in the absence of ions, they have any predictive capability of heterodimer stability. These observations extend to salts of SA, MSA and NA with nine bases: ammonia (AMM), methylamine (MA), dimethylamine (DMA), trimethylamine (TMA), trimethylamine N-oxide (TMAO), guanidine (GUA), monoethanolamine (MEA), putrescine (PUT) and piperazine (PZ) (Table 1). In addition to these molecular properties, we further explore the relationship between heterodimer stability and NPF rate for SA salts. The goal of this work is to develop computationally efficient approaches for calculating NPF rate that can be applied to models that estimate the impacts of NPF on climate and air quality. We represent NPF rate as J4×4, the rate at which a cluster larger than four acid and four base molecules is formed. A cluster of this size can range in diameter from 1 to 1.5 nm, depending on the constituent acid and base. We analyze the relationship between heterodimer stability and the theoretically predicted J4×4for sulfuric acid salts over a range of monomer concentrations from 105to 109molec cm−3and temperatures from 248 to 348 K. The concentration of heterodimers was calculated from heterodimer stability, temperature and monomer concentrations for the case where acid and base monomer concentrations are approximately equal. This results in a parametrization for J4×4as a function of heterodimer concentration that can be applied to any acid–base system. These results were compared to J1.7rates measured at the CLOUD (Cosmics Leaving OUtdoor Droplets) chamber for SA–AMM salts. We note that the relationship between J4×4and heterodimer concentration is not unique but depends on both temperature and monomer concentration. However, if the dependent variable is redefined as a term that we call the “normalized heterodimer concentration”, or 8, then a simple monotonic relationship develops that can be used to predict J4×4for weak salts of sulfuric acid, wherein their difference in gas-phase acidity (1GA) is greater than 95 kcal mol−1. We believe that this approach is generalizable to any acid–base system, allowing accurate predictions of NPF rates over a wide range Atmos. Chem. Phys., 21, 11637–11654, 2021 https://doi.org/10.5194/acp-21-11637-2021 S. Chee et al.: A predictive model for salt nanoparticle formation 11639 of monomer concentration, temperature and ambient pressure. 2 Computational methods Two-component acid–base particle formation was studied by making systematic changes in temperature and concentration to understand the effects of simulation conditions and acid/base molecular properties on J4×4. Correlations of J4×4 with different molecular properties provided insight into the critical factors of cluster formation. Properties listed in Table 2 were examined as possible variables that may have a role in stabilizing clusters and enhancing particle formation. 2.1 Cluster thermodynamics In order to simulate cluster formation and growth, one must calculate accurate structures and thermochemical properties of neutral SA–base clusters up to the cluster size of four SA and four base molecules (4SA4base). Thermochemistry of clusters containing AMM, DMA, GUA and TMAO were taken from our previous studies (Myllys et al., 2018, 2019b, 2020). Thermochemistry of clusters with MEA, PUT and PZ were taken from a database (Elm, 2019), collected from original publications of Xie et al. (2017), Elm et al. (2017) and Ma et al. (2019). Available structures with MA and TMA were taken from Olenius et al. (2017) and, to be consistent with the level of theory used, structures were optimized and frequencies calculated at the ωB97X-D/6- 31++G** level using rigid rotor–harmonic oscillator approximation, and electronic energies corrected at the DLPNO– CCSD(T)/aug-cc-pVTZ level with TightPNO, TightSCF and GRID4 keywords. In addition, for the missing structures, we performed a configurational sampling as explained in Kubeˇ cka et al. (2019). Briefly, to create the initial cluster structures, we used 3000 random guesses and 100 exploration loops, with a scout limit of four in the ABCluster program, and for each building block combination, we saved 300 of the lowest energy structures that were subsequently optimized by the tight-binding method GFN2-xTB with a very tight optimization criterion (Zhang and Dolg, 2016, 2015; Bannwarth et al., 2019). Based on the electronic energies, radius of gyration and dipole moments, we separated different conformers, which were then optimized using the ωB97X-D/6-31+G* level of theory. Based on the obtained electronic energies, we selected structures with a maximum of Nkcal mol−1from the lowest electronic energy (where Nis the number of molecules in the cluster). For the lowest free energy clusters, Gibbs free-binding energies were calculated at the DLPNO–CCSD(T)/aug-cc-pVTZ//ωB97X- D/6-31++G** level of theory (Riplinger and Neese, 2013; Riplinger et al., 2013; Myllys et al., 2016a, b). In addition of a full data set for SA–base clusters, we studied heterodimers of NA and MSA with the nine abovementioned bases. The same quantum chemical methods were used as in SA–base calculations. In order to detect whether proton transfer was occurring in the heterodimer, the Molden program (Schaftenaar and Noordik, 2000) was used to visualize the global minimum structure. Gas-phase basicity and proton affinity values were computed using the same level of theory. Gaussian 16 RevA.03 (Frisch et al., 2016) was used to optimize geometries and calculate vibrational frequencies, and Orca version 4.2.1 (Neese, 2012) was used for singlepoint energy corrections. 2.2 Particle formation simulations Theoretical methods allow us to perform particle formation simulations at conditions where particle formation rates are not experimentally measurable. This means that very low or high temperatures and vapor concentrations can be used to estimate J4×4. While some values in this range might not be directly “atmospherically relevant”, these calculations can lead to a deeper understanding of the non-linear behavior of nucleation as a function of vapor concentrations and/or temperature. It is also possible to study cluster formation of different compounds under identical conditions because there are no instrumental limitations or measurement biases. The calculated thermodynamic data sets for SA–base clusters were used as input in Atmospheric Cluster Dynamics Code (ACDC), the detailed theory of which is explained in McGrath et al. (2012). Briefly, the ACDC model simulated particle formation by solving the cluster distribution considering collision, evaporation and removal processes. The collision coefficients were computed from kinetic gas theory and the evaporation rates from quantum chemical Gibbs free energies assuming detailed balance. The model calculated the rate constants for each process among the population of clusters and vapor molecules and solved the discrete general dynamic equations for each cluster type. We have performed J4×4simulations at temperatures of 248–348 K using SA and base vapor concentrations of [acid] =[base]=105–109cm−3. Simulated J4×4values are given in the Supplement. Simulations were performed for neutral clustering pathways at dry conditions due to computational (quantum chemical) restrictions. It should be noted that the cluster sampling procedure were performed at 298 K, and those structures and thermodynamic data (1H and 1S) have been used in simulations at all temperatures. Thus, at lower or higher temperatures, slightly different global minimum structures might exist. Additionally, the simulation box size of four acid and four base molecules might be too small (i.e., critical cluster is outside of a box) at high temperatures and low concentrations. This leads overestimated particle formation rates, as discussed in Besel et al. (2020), where the effect of simulation settings was studied in the case of ammonia and sulfuric acid nucleation. https://doi.org/10.5194/acp-21-11637-2021 Atmos. Chem. Phys., 21, 11637–11654, 2021 11640 S. Chee et al.: A predictive model for salt nanoparticle formation Table 1. Acid and base compounds in this study. Abbreviations are as follows: ammonia (AMM), methylamine (MA), dimethylamine (DMA), trimethylamine (TMA), trimethylamine N-oxide (TMAO), guanidine (GUA), monoethanolamine (MEA), putrescine (PUT) and piperazine (PZ), sulfuric acid (SA), methanesulfonic acid (MSA) and nitric acid (NA). Table 2. Experimental and calculated properties examined in this study. Property Source Gas-phase acidity (GA)bcalculated in this work Difference between GA of an acid HA and a conjugate acid of a base BH+(1GA) calculated in this work Aqueous-phase acidity (pKa)bfrom Haynes (2014) Difference between pKaof HA and BH+(1pKa) from Haynes (2014) Vapor pressure from literaturea Electrochemical properties: dipole moment and polarizability calculated in this work Heterodimer stability (1Gheterodimer, free energy of a complex having one acid and one base) calculated in this work Remaining H-bond donors on base molecule in heterodimer inferred Proton transfer in heterodimer inferred aStull (1947), Aston et al. (1937, 1939), Swift and Hochanadel (1945), Matthews et al. (1950), EPISUITE v4.11. bAcidity of an acid HA or bases’ conjugate acid BH+. 3 Results and discussion 3.1 Heterodimer stability results In the cluster formation process, the changes in enthalpy (1H) and entropy (1S) are always negative because hydrogen bond formation is an exothermic process in which the degrees of freedom are decreasing when isolated molecules become one entity. Gibbs free energy is calculated from 1H and 1S as a function of temperature by 1G =1H −T 1S, (1) where 1G decreases as temperature decreases. Lower 1Gheterodimer values correspond to more stable heterodimers. However, while a negative 1Gheterodimer value indicates a spontaneous reaction in solution at standard conditions, heterodimer formation in the gas phase under atmospheric conditions also depends on the acid and base vapor concentrations. Table 3 presents enthalpies, entropies and Gibbs free energies of SA–base heterodimer formation at 298 K, and corresponding tables for MSA and NA are given in the Supplement. From these data, heterodimer stability can be calculated at other temperatures readily for all 27 salts studied here. Our calculated 1Gheterodimer value for SA–AMM indicates a less stable heterodimer than the SA–amines heterodimers, which is consistent with numerous other studies (Kurtén et al., 2008; Nadykto et al., 2011; Leverentz et al., 2013; Kupiainen et al., 2012). MA and MEA are the weakest heterodimer stabilizers among the amines; DMA, TMA and PZ are stronger and form approximately equally stable heterodimers. Of these nine bases, the most stable heterodimers are formed with TMAO, GUA and PUT. Atmos. Chem. Phys., 21, 11637–11654, 2021 https://doi.org/10.5194/acp-21-11637-2021 S. Chee et al.: A predictive model for salt nanoparticle formation 11641 Table 3. Calculated enthalpy (1Hheterodimer in kcal mol−1), entropy (1Sheterodimer in cal (mol K)−1) and Gibbs free energy (1Gheterodimer in kcal mol−1) for SA–base heterodimer formation at 298 K. BASE 1Hheterodimer 1Sheterodimer 1Gheterodimer AMM −15.1−29.2−6.4 MA −18.2−33.6−8.2 DMA −22.2−30.3−13.2 TMA −23.6−35.2−13.1 TMAO −32.2−34.9−21.8 GUA −29.4−30.4−20.3 MEA −21.8−38.2−10.4 PUT −28.9−44.8−15.6 PZ −22.8−33.3−12.9 The molecular structures of SA–base heterodimers are presented in Fig. 1 and for MSA and NA heterodimers in the Supplement. AMM is the only base which is unable to accept a proton from SA in the heterodimer structure; the heterodimer is held together via one hydrogen bond between AMM and SA. All other base compounds accept a proton from SA and form an ion pair with the deprotonated SA, bisulfate. Protonated TMA and TMAO form only one hydrogen bond with bisulfate, whereas the other bases form two hydrogen bonds. In the SA–PUT heterodimer, PUT also forms an intramolecular hydrogen bond via its protonated and non-protonated amino groups. 3.2 Molecular properties that affect heterodimer stability (1Gheterodimer) 3.2.1 Evaluation of gas-phase versus aqueous-phase acidity Figure 2 shows that gas-phase and aqueous-phase acidity values do not trend the same amongst the nine bases, wherein we define acidity of a base to be the acidity of the conjugate acid (i.e., the gas-phase or aqueous-phase acidity of NH3refers to the acidity of the conjugate acid, NH+ 4). For NR3com- pounds, where Ris either H or CH3, the gas-phase monomer acidities (GAs) directly follow the number of substitutions as AMM <MA <DMA <TMA. This means that when removing a proton from isolated gas-phase BH+compound, the Gibbs free reaction energy has the largest value in the case of TMA. That is because the methyl groups stabilize cation formation by distributing the charge. In the aqueousphase (pKa), however, the basicities have a different order: AMM <TMA <MA <DMA. This means that DMA has the largest proportion of protonated base cations in water solution. DMA has two methyl groups that facilitate protonation, and H-bond formation with water molecules provides additional stabilization. In the case of TMA, the hydration is very limited due to the steric hindrance of three methyl groups, and thus TMA has lower aqueous-phase basicity than DMA and MA. Because the basicity order of amines in the gas-phase directly follows the substitution order, the anomalous inversion of basicities in aqueous phase can be attributed to the stabilization effect of surrounding solvent molecules (Seybold and Shields, 2015). In the gas phase, the strongest bases are, in decreasing order, PUT, TMAO and GUA, whereas in the aqueous phase the order is GUA, PUT and DMA. GUA is a very strong base both in gas and aqueous phases because its cationic form has six πelectrons that are delocalized over the Y- shaped plane. This D3h-symmetric structure of guanidinium makes it extraordinarily stable. TMAO is very strong base in the gas phase because of its zwitterionic bond, where oxygen has a negative charge that strongly attracts H+. In the aqueous phase, polar solvent molecules are capable of stabilizing the zwitterionic bond in TMAO; thus, TMAO is the weakest base in the water solution. The reason why PUT is the strongest base in the gas phase is related to the change of its configuration between neutral and cationic forms. The neutral form of PUT is linear, but the cation is cyclic as the protonated and deprotonated amino groups are hydrogen bonded to each other as shown in Fig. 3. The Gibbs free energy difference between cyclic global minimum configuration and lowest acyclic local minimum configuration is 14.6 kcal mol−1, which is the additional stabilization caused by the H bond in gas phase. The gas basicity of PUT calculated based on the acyclic form would be 215.2 kcal mol−1, which is very close to that of DMA – and interestingly the pKavalues of DMA and PUT are very close to each other. This could indicate that protonated PUT is in aqueous phase mainly in its acyclic form and is stabilized by H bonds with water molecules in the same manner as DMA. As PUT and PZ are diamines, they can accept two protons and form baseH2+ 2cations. The PA and GA values for the second protonation reaction are significantly smaller than for the first protonation reaction: for PUT 130.6 and 125.2 kcal mol−1and for PZ 121.0 and 113.3 kcal mol−1, respectively. While the PA and GA values can be measured for the first protonation reaction for each base, there was no experimental data found for the second protonation reaction. Experimental PA and GA values from Hunter and Lias (1998) are given in the Supplement, and good agreement with our calculated values is shown. PA, GA and pKa values are listed for SA, MSA and NA in the Supplement. Because heterodimer stability has been shown to be a good proxy for J4×4, we have plotted the correlation between 1Gheterodimer and 1GA and 1pKato probe the hypothesis that acid and base strength predict the formation of the heterodimer (Fig. 4). Here, 1GA is defined to be the difference between the GA of the acid and the GA of the protonated base. And similarly the 1pKavalue is defined as the difference between the pKaof the acid and the pKaof the protonated base. All pKavalues were taken from literature as bulk aqueous-phase dissociation constants, whereas GA valhttps://doi.org/10.5194/acp-21-11637-2021 Atmos. Chem. Phys., 21, 11637–11654, 2021 11642 S. Chee et al.: A predictive model for salt nanoparticle formation Figure 1. Heterodimers of SA with AMM, MA, DMA, TMA, TMAO, GUA, MEA, PUT and PZ, respectively. Figure 2. Calculated GA vs. literature pKavalues from Haynes (2014). Figure 3. Cyclic and acyclic configurations of protonated PUT. The lowest energy acyclic structure is 14.6 kcal mol−1higher in free energy than the cyclic, global minimum structure. ues were calculated for this study. By definition, the larger the 1GA, the less favorable the acid–base reaction is in the gas phase. Similarly, the more positive the 1pKa, the less favorable the acid–base reaction is in the bulk aqueous phase. Over the observed 1GA, as 1GA increases, the less stable the heterodimer. The story is similar for 1pKa: as 1pKain- creases, the heterodimer becomes less stable. However, for Figure 4. Calculated 1GA and 1pKaplotted against 1Gheterodimer. Each data point represents an acid–base pair between either SA, NA or MSA with either AMM, MA, DMA, TMA, TMAO, GUA, MEA, PUT or PZ. Blue text represents 1GA values, while red text represents 1pKavalues. Text markers are centered over the data point. 1pKa, TMAO salts seem to deviate drastically from the trend. Indeed, this is most likely because TMAO is more able to be stabilized by water molecules in the bulk aqueous phase and its proton exchange in the gas phase is not well represented by pKa(Myllys et al., 2021). Otherwise, the trend of 1pKamatches up well with that of 1GA. These results demonstrate that acid and base strength have a clear Atmos. Chem. Phys., 21, 11637–11654, 2021 https://doi.org/10.5194/acp-21-11637-2021 S. Chee et al.: A predictive model for salt nanoparticle formation 11643 relationship with 1Gheterodimer and that 1GA can be used in parameterizations of 1Gheterodimer.1GA is even less computationally intensive than 1Gheterodimer because it only models the removal of a proton from the original molecule in comparison to modeling the interactions of two molecules. In addition, GA values can be calculated for an array of acids and bases to get 1GA for a larger combination of acids and bases rather than modeling 1Gheterodimer for each acid–base pair. For example, in this study, three acids and nine bases were studied: to calculate 1GA for all combinations, only 12 reactions need to be simulated; in contrast, 1Gheterodimer would need to be calculated for each of the 27 salts. Because the GA values calculated here agree well with those experimentally determined in Hunter and Lias (1998), this modeling approach may be a simpler, more consistent method to predict GA values for yet-unstudied bases, including those that are atmospherically relevant. Figure 5 illustrates how 1GA is a better predictor of proton transfer in the gas phase than 1pKa. In general, acid– base pairs with 1GA of 103 kcal mol−1or below undergo proton transfer, and thus 1GA provides a threshold for cluster formation. This is consistent with the stronger trends between heterodimer stability and GA than heterodimer stability and pKa, the latter of which was affected by the solubilities of the acids and bases, which is not relevant to cluster formation and growth in the gas phase. Interestingly, the NA–PZ salt is an anomaly in the cutoff for 1GA in predicting proton transfer, with 1GA value of 98.9 kcal mol−1, yet there is no proton transfer in the global minimum structures of heterodimer. However, there exists a local minimum structure in which proton transfer occurs that is only 1.8 kcal mol−1higher in free energy than the global minimum. Figure 6 shows that in the proton transferred form of the NA–PZ pair, the second H-bond formation, which is needed to stabilize the anion–cation pair, is unfavorable because of the induced ring strain. Generally, NA is less likely to form two H bonds with a base than SA or MSA as the angle of O–N–O is 120◦, whereas the O–S–O angle in SA and MSA are 109◦, and therefore the ring strain would be high in NA salts (with an exception for GUA as shown in the Supplement). Overall, heterodimer proton transfer only occurs in clusters with a 1GA smaller than 103 kcal mol−1 (NA–PUT) with the exception of NA–PZ. In general, this strengthens the idea that 1GA is a better estimate of gasphase reactivity than 1pKaand emphasizes the importance of using thermodynamic constants that accurately represent the systems being studied. 1GA and 1pKavalues can and should be used in lab settings to gauge the likelihood of nucleation. For example, numerous studies, including those in our own lab, show that oxalic acid does not form particles with any of the methylated amines (MA, DMA, TMA) in a two-component system at 298 K (Arquero et al., 2017). The most negative 1pKa value for these oxalic acid salts is −9.45, which is more positive than any of the systems studied here. Considering that NA–AMM does not form particles at room temperature even at high concentrations, its 1pKavalue of −10.7, or its 1GA value of 122.65 kcal mol−1, could be used as a benchmark for predicting particle formation at room temperature. This cutoff is dependent on both temperature and the concentrations of precursor acid and base and should be viewed as a qualitative means for predicting NPF at room temperature. A more accurate means of estimating NPF rates that accounts for both temperature and precursor concentration is presented in Sect. 3.3.1. It is worth of mentioning that our model for using heterodimer stability to predict particle formation rates is valid only for acid–base clusters and not for organic acid– inorganic acid clusters. Thus, for instance a formation free energy value of oxalic acid–sulfuric acid heterodimer cannot be used to predict particle formation efficiency using any formula presented in this paper. 3.2.2 Factors that do not affect heterodimer stability Figure 7 shows the relationship between base vapor pressure and heterodimer stability (1Gheterodimer), which is plotted to explore the hypothesis that the volatility of the base, which is typically much higher than that of the accompanying acid, is a limiting factor that drives NPF. The lack of correlation suggests that acid–base reactive uptake, leading to salt formation, is the dominant mechanism and that volatility of the constituent acid and base plays a relatively minor role in heterodimer stability. However, volatility plays a key role in cluster and nanoparticle growth, wherein low volatility compounds in the atmosphere (i.e., H2SO4) are still very important for understanding NPF. It is important to emphasize that this lack of correlation between vapor pressure and heterodimer stability is only observable because the bases have different structural properties. Otherwise, if only AMM, MA, DMA and TMA were studied, then trends for vapor pressure and heterodimer stability would follow the trend of the more volatile base making a less stable heterodimer, which is untrue. Since the most well-studied bases in the atmosphere are AMM, MA, DMA and TMA, due to their relative abundance and contribution to NPF, it may be tempting to make conclusions on base behavior in NPF based solely on those four bases. However, these correlations – or lack thereof – highlight the importance of a wider breadth of study for us to better understand how bases behave in the atmosphere. This disappearance of a trend as more bases are included applies to the dipole moment and polarizability of the base as well (see Supplement). However, it is worth noting that while base vapor pressure does not affect heterodimer stability, it may have a larger role in determining particle composition as particles grow to a size that represents bulk systems (Lawler et al., 2016; Chen and Finlayson-Pitts, 2017; Myllys et al., 2020; Chee et al., 2019). https://doi.org/10.5194/acp-21-11637-2021 Atmos. Chem. Phys., 21, 11637–11654, 2021 11644 S. Chee et al.: A predictive model for salt nanoparticle formation Figure 5. 1GA and 1pKavalues separated based on whether the heterodimer structure exhibits proton transfer. The dashed gray line on the 1GA graph at 103 kcal mol−1shows the cutoff point for proton transfer. Figure 6. Deprotonated (a) and protonated (b) conformers of PZ for the NA–PZ salt showing the ring strain necessary to form another intermolecular hydrogen bond. 3.3 Heterodimer stability vs. J4×4 The stabilities of a heterodimer and other small clusters are known to affect the ability of a cluster to grow to a large aerosol particle (Almeida et al., 2013; Elm, 2017; Olenius et al., 2013). We now correlate 1Gheterodimer with calculated J4×4for all nine bases with SA at varying conditions to observe the change in new particle formation rate over the temperature range of 248–348 K (Fig. 8a), and acid and base monomer concentrations from 105–109molec cm−3 (Fig. 8b). For reference, a Jof 0.1 cm−3s−1is also indicated, which can be viewed as a lower limit for observed atmospheric J4×4(Kerminen et al., 2018). We emphasize that these some of the concentrations and temperatures might not be very common in the atmosphere. However, through these systematic changes in temperature and concentrations, we are able to gain insight into the predictors of cluster formation and growth. Previously, theoretically calculated J4×4for reactions of SA with DMA or AMM have been shown to be a good approximation for experimentally determined NPF rates observed at the CLOUD chamber (Myllys et al., 2019b). As Fig. 8a shows, J4×4follows a lognormal relationship with 1Gheterodimer. This makes sense in that, for the most stable heterodimers like salts of TMAO and GUA, J4×4approaches the kinetic limit and simply cannot form any faster. However, as heterodimer stability decreases, the evaporation of a heterodimer occurs faster than its collision with vapor molecules or other clusters, which results in a reduction in J4×4. In contrast, when temperature is held constant and base concentration is varied (Fig. 8b), the fit curve shape remains the same as J4×4is shifted upwards with increasing starting concentrations until the kinetic limit is reached. The changing relationship between 1Gheterodimer and J4×4with varying temperature can be attributed to the change in the thermodynamics of the reaction, while the shift in NPF rate with respect to 1Gheterodimer with varying concentration can be attributed to Atmos. Chem. Phys., 21, 11637–11654, 2021 https://doi.org/10.5194/acp-21-11637-2021 S. Chee et al.: A predictive model for salt nanoparticle formation 11651 pole Electrostatics and Density-Dependent Dispersion Contributions, J. Chem. Theory Comput., 15, 1652–1671, https://doi.org/10.1021/acs.jctc.8b01176, 2019. Barsanti, K. C., McMurry, P. H., and Smith, J. N.: The potential contribution of organic salts to new particle growth, Atmos. Chem. 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