A predictive model for salt nanoparticle formation using heterodimer stability calculations

. 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 transfer-ence, 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 J 4 × 4 , the theoretically predicted formation rate of a four-acid, four-base cluster, for sulfuric acid salts over a range of monomer concentrations from 10 5 to 10 9 molec cm − 3 and temperatures from 248 to 348 K and found that heterodimer stability forms a lognormal relationship with J 4 × 4 . However, temperature and concentration effects made it difﬁcult to form a predictive expression of J 4 × 4 . In order to reduce those effects, heterodimer concentration was calculated from heterodimer stability and yielded an expression for predicting J 4 × 4 for 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 J 4 × 4 by deﬁning a new term that we call the normalized heterodimer concentration, (cid:56) . A plot of J 4 × 4 vs. (cid:56) collapses 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 J 4 × 4 within 2 orders of magnitude in atmospheric models.

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). 5 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 J 1.5 , the rate at which a cluster larger than 4 acid and 4 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 10 J 1.5 for sulfuric acid salts over a range of monomer concentrations from 10 5 to 10 9 molec cm −3 and 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 J 1.5 as a function of heterodimer concentration that can be applied to any acid-base system. These results were compared to J 1.7 rates measured at the CLOUD (Cosmics Leaving OUtdoor Droplets) chamber for sa-amm salts. We note that the relationship 15 between J 1.5 and 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 Φ, then a simple monotonic relationship develops that can be used to predict J 1.5 for all weak salts of sulfuric acid, a system that is of great interest in the atmosphere. We believe that this approach is generalizable to any acid-base system, allowing accurate predictions of NPF rates over a wide range of monomer concentration, temperature, and ambient pressure. 20 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 J 1.5 . Correlations of J 1.5 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. 25
In addition of a full data set for sa-base clusters, we studied heterodimers of na and msa with above-mentioned nine bases.

5
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)

Particle formation simulations
Theoretical methods allow us to perform particle formation simulations at any conditions. This means that very low or high temperatures and vapor concentrations can be used to estimate J 1.5 . 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 15 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), which detailed theory 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 model calculated the rate constants for each process among the population of clusters and vapor molecules and solved the discrete general dynamic 20 equations for each cluster type. We have performed J 1.5 simulations at temperatures of 248-348 K using sa and base vapor concentrations of [acid]=[base]=10 5 -10 9 cm −3 . Simulated J 1.5 values are given in the supporting information (SI). Simulations were performed for neutral clustering pathways at dry conditions due to computational (quantum chemical) restrictions.
3 Results and discussion

25
In the cluster formation process, the changes in enthalpy (∆H) and entropy (∆S) 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 ∆H and ∆S as a function of temperature by where ∆G decreases as temperature decreases. Lower ∆G heterodimer values correspond to more stable heterodimers. However, 30 while a negative ∆G heterodimer 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 SI. From these data, heterodimer stability can be calculated at other temperatures readily for all 27 salts studied here. Our calculated ∆G heterodimer 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., 5 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. The molecular structures of sa-base heterodimers are presented in Fig. 1 and for msa and na heterodimers in the SI. Amm is the only base which is unable to accept a proton from sa in the heterodimer structure; the heterodimer is held together via 10 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 (∆G heterodimer ) 15 3.2.1 Evaluation of gas-phase versus aqueous-phase acidity Figure 2 shows that gas phase and aqueous phase basicity values do not trend the same amongst the nine bases. For NR 3 compounds, where R is either H or CH 3 , the gas-phase monomer basicities 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 20 Figure 1. Heterodimers of sa with amm, ma, dma, tma, tmao, gua, mea, put and pz, respectively. the charge. In the aqueous-phase (pK a ), however, the basicities have an 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 5 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 phase because its cationic form has six π-electrons that are delocalized over the Y-shaped plane. This D 3h -symmetric structure of guanidinium makes it extraordinarily stable. Tmao 10 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 15 cyclic global minimum configuration and lowest acyclic local minimum configuration is 14.6 kcal/mol, 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, which is very close to that of dma -and interestingly the pK a values 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 baseH 2+ 2 cations. 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 and for 5 pz 121.0 and 113.3 kcal/mol, 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 SI and good agreement with our calculated values is shown. PA, GA and pK a values are listed for sa, msa and na in the SI.
Because heterodimer stability has been shown to be a good proxy for J 1.5 , we have plotted the correlation between ∆G heterodimer 10 and ∆GA and ∆pK a to probe the hypothesis that acid and base strength predict the formation of the heterodimer (Figure 4).
Here ∆GA is defined to be the difference between the GA of the acid and the GA of the protonated base. And similarly the ∆pK a value is defined as the difference between the pK a of the acid and the pK a of the protonated base. All pK a values were taken from literature as bulk aqueous phase dissociation constants, whereas GA values were calculated for this study. By definition, the larger the ∆GA, the less favorable the acid-base reaction is in the gas phase. Similarly, the more positive the ∆pK a , the less favorable the acid-base reaction is in the bulk aqueous phase. Over the observed ∆GA, as ∆GA increases, the less stable the heterodimer. The story is similar for ∆pK a : as ∆pK a increases, the heterodimer becomes less stable. However, for ∆pK a , tmao salts seem to deviate drastically from the trend.

5
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 pK a . Otherwise, the trend of ∆pK a matches up well with that of ∆GA. These results demonstrate that acid and base strength have a clear relationship with ∆G heterodimer , and that ∆GA can be used in parameterizations of ∆G heterodimer . ∆GA is even less computationally intensive than ∆G heterodimer because it only models the removal of a proton from the original molecule in comparison to modeling the interactions of two molecules. 10 In addition, GA values can be calculated for an array of acids and bases to get ∆GA for a larger combination of acids and bases rather than modelling ∆G heterodimer for each acid-base pair. For example, in this study, 3 acids and 9 bases were studied: to calculate ∆GA for all combinations, only 12 reactions need to be simulated; in contrast, ∆G heterodimer 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 yetunstudied bases, including those that are atmospherically relevant. Figure 5 illustrates how ∆GA is a better predictor of proton transfer in the gas phase than ∆pK a . In general, acid-base pairs with ∆GA of 103 kcal/mol or below undergo proton transfer, and thus ∆GA provides a threshold for cluster formation. This is consistent with the stronger trends between heterodimer stability and GA than heterodimer stability and pK a , the latter of 5 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 ∆GA in predicting proton transfer, with ∆GA value of 98.9 kcal/mol, 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 higher in free energy than the global minimum. Figure   10 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 SI). Overall, heterodimer proton transfer only occurs in clusters with a ∆GA smaller than 103 kcal/mol (na-put) with the exception of na-pz. In general, this strengthens the idea that 15 ∆GA is a better estimate of gas-phase reactivity than ∆pK a and emphasizes the importance of using thermodynamic constants that accurately represent the systems being studied.
∆GA and ∆pK a values 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 ∆pK a 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 ∆pK a value of −10.7, or its ∆GA value of 122.65 kcal/mol, 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.

5
A more accurate means of estimating NPF rates that accounts for both temperature and precursor concentration is presented in Section 3.3.1. Figure 7 shows the relationship between base vapor pressure and heterodimer stability (∆G heterodimer ), 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. 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 15 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 SI). However, it is worth noting that while base vapor pressure 20 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). 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;Olenius et al., 2013) We now correlate ∆G heterodimer with calculated J 1.5 for 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 (Figure 8a), and acid and base monomer concentrations from 10 5 -10 9 molec cm -3 (Figure 8b). For reference, a J 5 of 0.1 cm −3 s −1 is also indicated, which can be viewed as a lower limit for observed atmospheric J 1.5 (Kerminen et al., 2018).

Factors that do not affect heterodimer stability
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 J 1.5 for reactions of sa with dma or amm have been shown to be a good approximation 10 for experimentally determined NPF rates observed at the CLOUD chamber (Myllys et al., 2019b). As Figure 8a shows, J 1.5 follow a lognormal relationship with ∆G heterodimer . This makes sense in that, for the most stable heterodimers like salts of tmao and gua, J 1.5 approaches 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 J 1.5 . In contrast, when temperature is held constant and base concentration is varied (Figure 8b), the lognormal relationship 15 remains the same across all cases and only the maximum J 1.5 is shifted until the kinetic limit is reached. The changing relationship between ∆G heterodimer and J 1.5 with varying temperature can be attributed to the change in the thermodynamics of the reaction, while the shift in NPF rate with respect to ∆G heterodimer with varying concentration can be attributed to the relatively higher number of collisions in a shorter period of time. This behavior matches the relationship of J with temperature and concentration found in classical nucleation theory (Arstila et al., 1999;Trinkaus, 1983;Vehkamäki et al., 2002): where J is the nucleation rate, Z is a kinetic pre-factor, W is the work of formation of the critical nucleus, and p(1, 2) and W (1, 2) are number concentration and cluster formation energy, respectively. Concentration is directly proportional to J, 5 whereas temperature contributes from within the exponential expression, which matches the behavior seen in Figure 8.
Interestingly, as temperature increases, this lognormal relationship transitions to linear, with a larger spread of data points around the trendline. Practically, this implies that ∆G heterodimer predicts theoretical J 1.5 well at cold temperatures, but additional factors become more prominent at warmer temperatures. To understand what processes are important for J 1.5 , we scaled the color on each of the bases to the number of hydrogen bond donors (HBD) remaining on the heterodimer after the proton was  With respect to the lognormal relationship between J 1.5 and ∆G heterodimer , tma and tmao, and to a lesser extent, dma, are below the trendline, and they have 0-1 remaining HBD. In contrast, amm, ma, mea, pz, and put have 2-4 remaining HBD and are closest to the trendline. Gua is the only molecule that has 5 remaining HBD, and consistently has a higher NPF rate than the trendline suggests. This behavior can be attributed to cluster growth being slightly dependent on how well the next molecules can "stick" onto the existing cluster, where if there are more remaining HBD on a cluster, it is easier and faster for the cluster 5 to grow. It is interesting that ma has higher NPF rates than the trendline compared to mea, put, and pz despite having either the same or one fewer HBD, but this may be attributed to the bulkiness of the alkyl groups attached to those amines, which may block the remaining HBD from participating in stabilizing the growing cluster.
These findings are notable in that ∆G heterodimer trends consistently with J 1.5 and deviations from these trendlines can be attributed to structural differences in the base, where a base with more HBD available on the heterodimer would have higher 10 predicted NPF rates than the trendline, with the inverse also being true. However, ∆G heterodimer varies strongly with temperature and concentration as described above, and as such is not conducive to predicting J 1.5 , which we attempt to remedy in the following two sections.
3.3.1 A generalized parameterization to predict J 1.5 In order to combine simulated particle formation rates at different conditions for all acid-base systems, we calculated the 15 heterodimer concentration, which is a function of ∆G heterodimer , temperature, and the concentration of the gaseous acid and base monomers. The stability of a heterodimer defines its theoretical maximum concentration at given conditions assuming the system is at equilibrium. Assuming mass balance for the heterodimer formation reaction leads to the following concentration under equilibrium conditions: The  We calculated heterodimer concentrations for CLOUD data whose acid and base concentrations were within 50% of each other according to Equation 3. All CLOUD data points were collected at temperatures of either 248 or 273 K (colored circles corresponding to color scale) and with monomer concentrations between approx. 10 8 -10 9 cm -3 . Figure 10a shows the temperature and concentration effects on heterodimer concentration for sa-amm salts. As one would 10 expect from Equation 3, as concentration increases, heterodimer concentration increases by two orders of magnitude (as reflected in the [heterodimer] term). However, because temperature affects both the calculation of ∆G heterodimer and heterodimer concentration, this relationship is not as simple. In general, as temperature decreases, heterodimer concentration increases.
As heterodimer concentration increases and temperature decreases, J 1.5 also increases, though we begin to see J 1.5 begin to saturate at 248 K and 10 9 cm -3 . Through the use of heterodimer concentration, we have been able to combine the two factors, temperature and monomer concentration, into one term, where we can now use it to compare (or predict) J 1.5 .
To test the robustness of our calculations, heterodimer concentrations of CLOUD experiments were calculated using Equation 3 and this study's calculated ∆G heterodimer values to compare our J 1.5 calculations to CLOUD's measured J 1.7 (Kirkby et al., 2011). Because heterodimer concentration can only be calculated for experiments run at approximately equal acid and 5 base concentrations, all experiments that had more than a 50% difference between monomer concentrations were excluded.
Twenty-one measured J 1.7 values met this criterion and are shown as filled circles in Figure 10. When using the closest temperature trendlines (i.e., CLOUD data measured at 273 K was compared to 278 K model trend) to predict the CLOUD data, the difference between the predicted and measured J were within 2 orders of magnitude. On the other hand, if concentration trendlines were used to predict J (i.e, CLOUD vapor concentrations were near 10 8 molec cm -3 so the modelled 10 8 molec 10 cm -3 trendline was used), differences of up to 4 orders of magnitude occurred. Trendline equations for sa-amm are shown in the SI, as well as difference plots to show the accuracy of the trendlines as discussed. Figure 11. Heterodimer concentration plotted against J 1.5 , wherein a) all data are represented with black dots, and b) data points are colored according to temperature and sized to reflect monomer concentrations (10 5 -10 9 cm -3 ). Data were fitted to an exponential function, which can be found in Equation 4.
All data calculated for this study are plotted in Figure 11, which spans 100 K and 5 orders of magnitude in monomer concentrations. Indeed, concentration and temperature effects are minimized compared to the direct comparison between J 1.5 and ∆G heterodimer (Figure 8). Because more J 1.5 were calculated for amm and gua, data points were left as black points to avoid complicating the data. Data were fitted to give the following equation: which can be used as a generalized equation to predict J 1.5 for acid-base particle formation at any (atmospheric relevant) conditions, given a calculated ∆G heterodimer and temperature and concentration. Because ∆G heterodimer requires significantly less computational power to calculate than J 1.5 , this trendline provides a method to quickly approximate J 1.5 .
Since the heterodimer concentration is still affected by changes in temperature and concentration, Equation 4 is only able to approximate J 1.5 to within 10 orders of magnitude. This is because of the large range of temperatures and concentrations calculated in this study, where, in general, for concentrations less than 10 7 cm -3 and temperatures greater than 298 K, predicted J 1.5 are below the trendline. Similarly, for concentrations more than 10 7 cm -3 and temperatures greater than 298 K, predicted J 1.5 are above the trendline, which can be seen in Figure 11b. 10 Though the 10 orders of magnitude uncertainty is large, Pierce and Adams (2009) have shown that 6 orders of magnitude uncertainty in new particle formation events in the atmosphere only contributed to a difference of 17% in modeled concentrations of cloud condensation nuclei (CCN) in the troposphere. Considering the simplicity of this calculation, this approach may improve estimates of global CCN in models that are limited by the computational expense of calculating J 1.5 .
3.4 System-specific parameterization for weak bases using normalized heterodimer concentration (Φ)

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Here we attempt to reduce this uncertainty for nine salts of sa and further simplify the expression used to calculate J 1.5 . We accomplish this by incorporating heterodimer concentration and monomer concentrations into a new independent variable, the normalized heterodimer concentration, Φ: When applied to ammonia, a simple monotonic relationship between Φ and J 1.5 becomes immediately apparent ( Figure   20 12a). Here we observe that temperature affects the value of Φ minimally, and that the effects of temperature and concentration are incorporated in the dependent variable resulting in relatively minor data spread. Again, CLOUD Φ values were calculated for comparison, and CLOUD data are all predicted within 2 orders of magnitude of the best exponential fit to the data (fit equation available in the SI). The dispersion in J 1.5 remains constant over all conditions explored.
As a contrast to the sa-amm system, we also examined the behavior of the sa-gua salt, a strong-acid and strong-base 25 combination. Figure 12b shows that a monotonic relationship does not apply for such systems. In fact, at each concentration, J 1.5 quickly reaches the kinetic limit and remains constant with temperature once monomer concentrations are above 10 7 cm -3 . Gua is likely insensitive to changes in temperature because gua is a strong base and forms more stable growing clusters than those of ammonia. In addition, at higher concentrations than 10 7 cm -3 , collisions are occurring so quickly that if the cluster evaporates a monomer, another monomer is able to readily take its place. In this way, gua salt J 1.5 are largely dictated 30 by monomer concentration rather than temperature.   When Φ is compared to J 1.5 for all bases (Figure 13a), we can immediately see that, in general, each base follows a unique trendline. Additionally, more bases follow the more monotonic behavior of sa-amm than sa-gua and increase in the data dispersion follows increasing basicity. This is apparent when each of the base datapoints are colored according to their ∆GA values (Figure 13b). In general, the larger ∆GA values correspond to more linear, less dispersed relationships between J 1.5 and Φ, and as ∆GA decreases, J 1.5 begin to saturate and dispersion increases. This change in behavior seems occur most dramatically as ∆GA decreases below 90 kcal/mol for the conditions shown here; however, it is likely for larger concentrations or lower temperatures, even the weakest of bases will saturate. The fact that ∆GA is directly linked to J 1.5 saturation highlights 5 how acid and base strength are crucial to understanding cluster formation and growth into particles.
Here, Φ can be used to predict J 1.5 relatively accurately for specific bases, as demonstrated by the CLOUD J 1.7 observations.
However, for bases with ∆GA below approximately 90 kcal/mol, prediction becomes more uncertain as the kinetic limit becomes easier to reach. This ∆GA cutoff of 90 kcal/mol means that the most abundant bases in the atmosphere, amm, ma, dma, and tma, are not expected to saturate in this model under atmospheric conditions and thus their J 1.5 can be approximated 10 relatively accurately using the results of this study. While this can only be used for experiments with acid and base monomer concentrations within 50% of each other over the concentrations and temperatures studied, this is a powerful predictive tool using only the term, Φ, which only requires the calculation of one computational parameter, ∆G heterodimer .
Because each base has its own correlation between Φ and J 1.5 , the trendlines here cannot be generalized to bases that are not described. For those bases not described here, Equation 4 should be used to approximate J 1.5 to within 10 orders of magnitude.

Conclusions
Here we have shown that heterodimer stability is largely predicted by the gas-phase acidity of the constituent acid and base across 27 acid-base pairs. In addition, we found that trends between heterodimer stability and physical properties such as volatility, dipole moment, and polarizability did not hold for the wide variety of bases studied here, despite a trend existing for the smaller set of amm, ma, dma, and tma. We emphasize here the importance of studying a variety of bases with different 20 structures and physical properties in order to make sure our understanding of salt NPF remains unbiased. We have also shown the relationship between J 1.5 and heterodimer stability and how it was affected by temperature and concentration. We show that deviations from the lognormal relationship were attributed to the remaining HBD available on the base molecule on the heterodimer. Then in order to devise a simple model to predict J 1.5 , we calculated heterodimer concentration from our heterodimer stability values. The effects of temperature and concentration on heterodimer concentration were much less than 25 that of those on ∆G heterodimer but still were present, as shown by the 25 different calculations of sa-amm J 1.5 . When compared to CLOUD experimental J 1.7 data, the sa-amm trendlines were able to predict J 1.5 within two orders of magnitude when the closest temperature trendline was used. We found that heterodimer concentration can be parameterized into a expression that can predict J 1.5 . Because of this, the more difficult to calculate parameter of J 1.5 could be replaced by the more easily acquired parameter of heterodimer stability. In addition, we have calculated a new parameter, the normalized heterodimer concentration,

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Φ, which minimized the effects of temperature and concentration even more than that of heterodimer concentration. We found that Φ reduces the complexity of calculating J 1.5 by producing a single, monotonic trendline for sa-amm, instead of 10 as it was for our calculations using heterodimer concentration as the independent variable. The ability of Φ to accurately predict J 1.5 applies to sa salts of weaker bases, as stronger bases quickly saturated to reach the kinetic limit. This behavior was exhibited more strongly for salts that had a ∆GA value smaller than 90 kcal/mol.
In addition, we have presented a facile way of predicting J 1.5 to within 10 orders of magnitude for salts of sa using a generalized parameterization (Equation 4). We also present a method to more accurately predict J 1.5 using the new parameter Φ for the nine sa salts studied here. It is important to note that, due to computational restrictions, all particle formation simulations 5 are performed for two-component neutral clusters with an absence of relative humidity. Thus theoretical results might vary compared to measured particle formation under atmospheric or laboratory conditions. Water enhancement of NPF is known to be greater with more available hydrogen bonding sites as shown in Yang et al. (2018), which may enhance the deviation from the lognormal relationship that was attributed to remaining HBD on the heterodimer. The enhancing effect of ions on the NPF rate can be several orders of magnitude for systems where small neutral clusters are unstable (e.g., ammonium salts in 10 this study), but is negligible with more stable clusters, like a strong acid and base pair (Myllys et al., 2019b). In addition, when more than two components are present at the same time in the atmosphere or even as a contaminant in laboratory, NPF can be largely enhanced due to synergistic effects (Glasoe et al., 2015;Jen et al., 2014;Yu et al., 2012;Temelso et al., 2018;Myllys et al., 2019a). It is infeasible to explicitly study of all possible combinations of multi-component acid and base mixtures, but perhaps in the future the synergy between different compounds and the role of water vapor could be estimated using some