Reducing chemical complexity in representation of new-particle formation: evaluation of simplification approaches

Abstract

Adequate representation of new-particle formation from vapors in aerosol microphysics and atmospheric transport models is essential for providing reliable predictions of ambient particle numbers and for interpretation of observations. Atmospheric particle formation processes involve multiple species, which complicates the derivation and implementation of data and parameterizations to describe the processes. Ideally, the representation of multi-compound mechanisms should be reduced, but remain accurate. Here, we evaluate common approaches to simplify the description of representative multi-compound acid–base chemistries by applying different theoretical molecular cluster data sets, focusing on simplifications (1) in formation rates that are used as input in aerosol process models and large-scale models, and (2) in cluster models that are applied to assess particle formation dynamics and survival to larger sizes. We test the following approaches: assuming non-interactive additive formation pathways, lumping of similar species, and application of quasi-unary approximations. We assess the possible biases of the simplifications for different types of chemistries and propose best practices for reducing the chemical complexity. We demonstrate that simplifications in formation rates are most often justifiable, but the choice of the preferred simplification method depends on the types of species and their similarity. Simplifications in cluster growth dynamics by quasi-unary approaches, on the other hand, are reasonable mainly for strong cluster formation involving very low-evaporating species and at excess concentration of the implicitly treated stabilizing compound.

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Environmental significance

New-particle formation is a major source of atmospheric aerosol particles, and thus an important factor affecting aerosol–cloud–climate processes. While model representation of particle formation mechanisms should preferably be concise and robust, the participation of multiple chemical species complicates the description. This work evaluates common approaches to simplify multi-compound chemistries, focusing on sulfuric acid–base systems. We provide assessments of reasonable simplifications for (1) incorporating formation rates in atmospheric models, and (2) detailed studies of new-particle formation and growth dynamics. This contributes to finding the best approaches for representing secondary particle sources, and mapping and reducing model uncertainties.

1 Introduction

Formation of new aerosol particles from condensable vapors makes a major contribution to atmospheric aerosol numbers, and has significant—but uncertain—effects on aerosol–cloud–climate interactions, the global radiation budget, and air quality.^1–3 The initial formation of nanometer-sized particles, occurring through clustering of gas-phase molecules to molecular clusters, is affected by multiple chemical compounds, including acid and base species and potentially also oxidized organic compounds.^4,5 The clustering can proceed through multi-compound chemical mechanisms or parallel pathways,^6,7 which need to be adequately represented in atmospheric aerosol models in order to capture the effects on particle concentrations.

The main drivers of the initial particle formation process include mixtures of acid and base compounds.^8–10 Specifically, sulfuric acid (H₂SO₄) is a central particle formation agent together with stabilizing base species, the most abundant of which is ammonia (NH₃). In addition, atmospheric amines efficiently form molecular clusters with H₂SO₄.¹¹ These include monoamines, i.e. mono-, di- and trimethylamine,¹² of which dimethylamine has often been studied as a representative amine species,¹³ as well as diamines.¹⁴ Amines are typically more efficient clustering agents than NH₃ but their atmospheric concentrations are generally much lower than that of NH₃, although elevated close to the sources.^15–17 Depending on the ambient conditions, these different base species can contribute to H₂SO₄-driven particle formation either alone or together.

The presence of multi-compound formation pathways increases the complexity of the description of the formation process in several ways. The functional form of the formation rate rapidly becomes very complex as the number of independent parameters increases, which complicates both the experimental and theoretical efforts to study the formation mechanisms, as well as their implementation in aerosol modeling. Experimental assessment of formation rates over a wide range of combinations of precursor vapors at different concentrations,¹⁸ as well as deriving robust parameterizations of the data, is laborious. Theoretical predictions of the properties of multi-compound molecular clusters of various compositions are likewise demanding. Such predictions are obtained by quantum chemical calculations,¹⁹ which require substantial computational resources when performed over large sets of cluster compositions. Simulations of molecular cluster population dynamics,^20,21 generally applied together with quantum chemical input data to yield cluster concentrations and formation rates, also become more complex. A very large number of different molecular compositions can both increase the simulation time and make interpretation of results less straight-forward. Finally, implementation of formation rates in atmospheric models requires more sophisticated interfaces when multiple data sets are needed to determine the rates for different vapor species and their combinations. Overall, reducing chemical complexity in the representation of nanoparticle formation mechanisms by appropriate simplifications is highly desirable

(1) due to limitations in available experimental and theoretical data,

(2) for an optimized description of the process in large-scale models, and

(3) for robust interpretation of measurements by molecular cluster dynamics modeling.

A common simplification that is routinely applied to account for several particle formation pathways is to assume that the formation rates J determined for different chemical systems are additive.^3,22 For instance, if there exist data for particle formation from compounds A and B, and from A and C, the rates are summed to yield the total formation rate as J_tot = J_AB + J_AC. This corresponds to assuming that clustering pathways AB and AC do not interfere, meaning that compounds B and C compete to form clusters with A and/or do not form bonds with each other. While a simplified treatment is unavoidable when there exist no data for the three-component system ABC, potential errors arising from the sum approach have not been assessed for typical atmospheric clustering agents.

Another approach to reduce the number of compounds is to approximate chemically similar compounds as a lumped species. This has been assumed, for example, for amines.²³ If the chemical properties of clusters in systems AB, AC, and ABC are very similar, it is justifiable to approximate the ABC system as AD, where D is a representative species with a concentration of [D] = [B] + [C]. However, in this case the sum approach may give distorted results, as it cannot generally be assumed that J_AD([D] = [D]₁ + [D]₂) = J_AD([D] = [D]₁) + J_AD([D] = [D]₂). That is, the choice of simplification should depend on the properties of the chemical systems.

Multi-compound systems can also be reduced to quasi-unary systems by implicit treatment of compounds.^24–28 This corresponds to assuming that one or more compounds do not need to be explicitly included, but only affect the cluster properties, most importantly the cluster evaporation rates which generally strongly depend on the molecular composition.^29,30 This corresponds to representing, for example, all cluster compositions A₃B_X that consist of three molecules of compound A and any number of molecules of compound B, as a single cluster A₃ with effective properties that are affected by the presence of B. Here, B is an enhancing species that cannot drive clustering on its own. This is justifiable, for instance, when the rate constants for collisions and evaporations of compound B are significantly higher than for A. The dynamic processes related to B thus occur in such short time scales that the cluster distribution is always instantaneously equilibrated with respect to B.^25,26 Representative properties may also be a justifiable assumption when the clusters are likely to exist in specific, thermodynamically favorable compositions. For example, H₂SO₄–dimethylamine clusters have been observed to favor a close to 1 : 1 molar ratio^27,30 and have thus been approximated as a quasi-unary compound consisting of H₂SO₄–dimethylamine pairs.^10,27

In this work, we evaluate the approaches to simplify the representation of multi-compound new-particle formation. We test the validity of the typical simplifications by comparison against explicit simulations for atmospherically relevant H₂SO₄–base chemistries. The purpose is to answer the following questions:

(1) Can the simplified approaches be expected to represent the multi-compound systems with reasonable accuracy?

(2) How significant errors may the simplifications cause, in terms of the magnitude and direction of the possible errors?

(3) Does the applicability of the simplifications depend on the properties of the chemical system and/or the ambient conditions?

2 Validation of simplifications against explicit simulations

The simplification approaches, depicted in Fig. 1, are validated against explicit simulations of representative multi-compound chemical systems over a wide range of atmospheric conditions. We apply available sets of quantum chemical data for the thermodynamic properties of H₂SO₄–base clusters together with the cluster population dynamics model ACDC,³¹ as detailed below.

Fig. 1 Schematic presentation of the approaches to simplify multi-component chemistries for (a) assessments of particle formation rate in the presence of multiple species, and (b) modeling of cluster growth by a quasi-unary framework. The axes depict cluster compositions as numbers of molecules of different species.

2.1 Chemical systems and thermodynamic data sets

We focus on H₂SO₄–base chemistries where the primary bases of interest include ammonia (NH₃) and dimethylamine (DMA). In order to ensure the generalizability of the results, we apply several quantum chemical data sets, as listed in Table 1. These are the most complete data sets available in terms of included cluster sizes and compositions, and they have been evaluated against laboratory measurements in previous works (see references in Table 1). These widely studied chemical systems represent weaker (NH₃) and stronger (DMA) clustering agents that are also expected to interact and exhibit synergistic effects.^32–34

Table 1 Chemical systems and quantum chemical data sets applied in this work. Data sets “RICC2” and “DLPNO” are computed at RICC2/aug-cc-pV(T+d)Z//B3LYP/CBSB7 and DLPNO-CCSD(T)/aug-cc-pVTZ//ωB97X-D/6-31++G(d, p) levels of theory, respectively. Different data sets also involve different configurational sampling approaches and theoretical approximations (see the ESI for more details)

Chemical system	Quantum chemical data sets	Cluster size range (max. numbers of H2SO4 and base molecules)	References
a For comparisons with H2SO4–NH3–DMA chemistry (Section 2.3.1), a smaller cluster set with up to 4 H2SO4 and 4 NH3 molecules is used for consistency. The 5 H2SO4, 5 NH3 set is used in the quasi-unary simulations (Section 2.3.2). b DLPNO_Besel is primarily applied for the H2SO4–NH3 chemistry in the quasi-unary simulations (Section 2.3.2) as it covers larger cluster sizes than other DLPNO-based data, which are used for the particle formation rate assessments (Section 2.3.1).
H2SO4–NH3, H2SO4–DMA, H2SO4–NH3–DMA	RICC2	4 H2SO4, 4 base, 5 H2SO4, 5 NH3^a	Almeida et al. (2013)^13
			Olenius et al. (2013)^20
			Kürten et al. (2016)^39
	DLPNO_Myllys	4 H2SO4, 4 base	Myllys et al. (2019a)^30
			Myllys et al. (2019b)^32
	DLPNO_Besel^b	6 H2SO4, 6 NH3	Besel et al. (2020)^40
	DLPNO_Kubečka	4 H2SO4, 4 base	Kubečka et al. (2022)^37
H2SO4–GUA	DLPNO_Myllys	4 H2SO4, 4 base	Myllys et al. (2018)^35
			Myllys et al. (2019a)^30
H2SO4–base1–base2 with all combinations of NH3, MA, DMA, TMA, EDA	DLPNO_Kubečka	4 H2SO4, 4 base	Kubečka et al. (2022)^37

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In addition, we perform tests with H₂SO₄–guanidine data. Guanidine, here referred to as GUA, is a very strong organobase that is able to form extremely low-evaporating clusters with H₂SO₄.^30,35,36 Finally, we apply a set of multi-base data to simulate H₂SO₄–base₁–base₂ chemistries with different combinations of bases, including NH₃, DMA, methylamine (MA), trimethylamine (TMA), and ethylenediamine (EDA).³⁷ These amine species are observed in the ambient atmosphere, and cover weaker (MA) and stronger (DMA, TMA, EDA) clustering strengths.^14,38 Further details of all data sets and simulation settings can be found in the ESI.†

2.2 Cluster dynamics simulations

Cluster concentrations and formation rates are simulated by the cluster population model ACDC,^20,31 which generates and solves the time derivatives of all cluster concentrations C_i for a given set of cluster compositions i:

In eqn (1), the first sum includes all collisions that create composition i, and the corresponding reverse evaporations. The second sum includes collisions of cluster i with other clusters, and the reverse evaporations that produce composition i. β_i,i is the rate constant for a collision between clusters i and j, and γ_i→j,i−j is the rate constant for evaporation of cluster i into products j and i − j. S_iC_i is a loss term corresponding to scavenging of clusters. Here, the collision constants are calculated using the kinetic gas theory, and the evaporation constants are obtained from the quantum chemical formation free energies.²⁰ The loss rate constants correspond to scavenging by larger aerosol particles, characterized by the H₂SO₄ condensation sink CS as S_i = CS × (d_p,i/d_p,H2SO4)^−1.6, where d_p,i and d_p,H2SO4 are the diameters of cluster i and H₂SO₄, respectively.⁴¹ Ionic molecules are included when available (see ESI Section 1†) by adding collisions with ionizing species in eqn (1), corresponding to ionization and recombination processes.²⁰

The solution to eqn (1) gives the cluster concentrations C_i, and the particle formation rate J is defined as

corresponding to the rate at which stable clusters grow beyond the simulated size range. In eqn (2), the summation goes over all collisions that lead to a stable collision product i + j according to the given stability criteria, denoted by {i + j | condition}²⁰ (Table S1†). We focus on steady-state conditions, and perform simulations over sets of atmospherically relevant vapor concentrations, temperatures, and CS, from clean to highly polluted environments. As H₂SO₄ molecules tend to cluster with strong bases, we define the H₂SO₄ vapor concentration as the total concentration of H₂SO₄ monomers and H₂SO₄·base, where “base” refers to any base molecule. This is consistent with measurements, as the clustered molecules are expected to contribute to the H₂SO₄ signal.⁴²

2.3 Simplified set-ups and comparison to explicit results

2.3.1 Representation of the total particle formation rate in atmospheric models. First, the H₂SO₄–NH₃–DMA data sets are applied to simulate a multi-compound system with chemically different but possibly interacting species with weaker and stronger cluster formation potentials. The assumption of non-interacting particle formation pathways is tested by comparing the three-compound formation rate J_3-comp to the sum of the formation rates ∑J_2-comp for the two-compound systems H₂SO₄–NH₃ and H₂SO₄–DMA (Fig. 1a). Second, ideal cases of chemically similar systems are constructed by applying the same thermochemical data for two separate simulation systems H₂SO₄–base₁ and H₂SO₄–base₂, where both base₁ and base₂ have the properties of either NH₃ or DMA. In this case, the three-compound system H₂SO₄–base₁–base₂ is correctly described by the lumping approach, that is, assuming only one base with a concentration of [base₁] + [base₂]. However, the sum approach may yield erroneous results. Finally, the DLPNO_Kubečka multi-base data set is applied to test if other base combinations behave similarly to the H₂SO₄–NH₃–DMA case, and to examine if there are different bases within the data set that can be lumped with reasonable accuracy.

The test cases assess the effects of the sum assumption for particle formation mechanisms that involve interacting species that are either (1) different, or (2) very similar in terms of their clustering efficiency. This addresses the question of potential errors in total particle formation rates in aerosol microphysics and large-scale atmospheric models, which generally apply the sum approach. Furthermore, tests with the DLPNO_Kubečka data provide assessments on whether relatively similar bases are better represented by the sum or the lumping approach.

2.3.2 Representation of cluster concentrations for assessments of formation and growth dynamics. Next, approaches to describe the cluster size distribution through a quasi-unary model are tested (Fig. 1b). First, the systems are reduced to a quasi-unary H₂SO₄ system by assuming that H₂SO₄-containing clusters are equilibrated with respect to the base species. That is, each n-mer (H₂SO₄)_n, n = 1, 2, 3, … exists in its equilibrium distribution for which the relative concentration of an n-mer containing m base molecules is⁴³where the summation goes over all possible numbers m_n of base molecules in the (H₂SO₄)_n cluster, ΔG_m is the binding free energy of the cluster with m base molecules, P_ref is the pressure at which the binding free energies are calculated, and P_base is the partial pressure of the base vapor. k_B and T are the Boltzmann constant and temperature, respectively. Effective collision and evaporation rate constants between n-mers i and j become²⁵andIn eqn (4) and (5), m_i and m_j are the numbers of base molecules in clusters i and j, respectively, and f_i,mi is the relative fraction of the composition with m_i bases in the total concentration of cluster i (eqn (3)). β_i,mi;j,mj is the collision rate of composition (i, m_i) with composition (j, m_j), and γ_{i+j,mi+mj→i,mi;j,mj} is the evaporation rate of (i + j, m_i + m_j) into (i, m_i) and (j, m_j). The equilibrium assumption is generally applied for water, for which the rate constants are typically orders of magnitude higher than for other species. Applicability on other stabilizing agents depends on whether the time scale of equilibration is sufficiently fast so that the distribution of different base contents m depends only on the thermodynamics (eqn (3)). The effects of the exact distribution on the effective collision constants β are typically minor, while the effective evaporation constants γ may be very sensitive to it.²⁵

Second, a more simplified approach for a quasi-unary substance is applied by assuming that the base content of a given n-mer corresponds to the most stable composition. The substance is assumed to consist of H₂SO₄·base heterodimers, and cluster evaporation rates are set to those corresponding to the lowest total evaporation rate for a given n (ESI Section 2.2†). In this case, it is assumed that the base content can be represented by a single composition, which further simplifies the model treatment and interpretation of results.

The (H₂SO₄)_n concentrations of the quasi-unary systems are compared to the total concentrations ∑(H₂SO₄)_n·(base)_m of the explicit systems for each n-mer. Here, the cluster concentrations are compared instead of formation rates as such one-compound model substances are typically applied to interpret observations of nanoparticle formation and growth^27,44 and to assess cluster survival to larger sizes.⁴⁵

3 Results and discussion

3.1 Simplifications in formation rates by assuming non-interacting or lumped species

The ratio of the summed formation rates ∑J_2-comp to the explicit three-component rate J_3-comp for H₂SO₄–NH₃–DMA at typical atmospheric base concentrations (Table 2) is shown in Fig. 2. The results are here shown only for the DLPNO_Myllys input data, as the RICC2 and DLPNO_Kubečka data give similar trends (see below). For such chemistry, the omission of base synergy can cause underprediction, but the errors are not very significant: the largest bias occurs at the combination of high [NH₃] (mainly above ∼10²–10³ ppt) and low [DMA] (mainly below a few ppt) when the bases contribute together in an optimal way by enhanced initial cluster stabilization (DMA) and enhanced cluster growth (NH₃). Under other conditions, [NH₃] is not high enough for significant enhancement, and/or the formation process becomes dominated by the stronger base DMA. While the high [NH₃]–low [DMA] conditions are common due to the different sources of the bases, the underprediction is typically clearly less than a factor of ∼10 at an average atmospheric scavenging sink of CS = 10⁻³ s⁻¹. The bias increases with CS, and is here up to a factor of ∼30 under polluted conditions with CS = 10⁻² s⁻¹ at which efficient growth becomes more critical.

Table 2 Typical atmospheric concentration ranges of NH₃ and amines from measurements

Base species	Clean and average environments (e.g. remote and marine sites)	Polluted environments (e.g. agricultural and urban sites, megacities)	References
NH3	Tens of ppt to hundreds of ppt	∼1 to ≳10 ppb	Nair and Yu (2020)^46
			Croft et al. (2016)^47
			Makkonen et al. (2014)^48
			Xiao et al. (2021)^9
Amines (MA, DMA, TMA)	<1 ppt	∼1 to ≳10 ppt	Yu and Luo (2014)^38
			Yao et al. (2018)^10

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Fig. 2 Comparison of simulated particle formation rates assuming non-interacting base species (H₂SO₄–NH₃ and H₂SO₄–DMA; ∑J_2-comp = J(AN) + J(AD)) or the fully interacting three-component H₂SO₄–NH₃–DMA system (J_3-comp = J(AND)). Ratios of ∑J_2-comp/J_3-comp at different concentrations of NH₃ and DMA at two different [H₂SO₄] and CS of 10⁻³ s⁻¹ (two left columns) and 10⁻² s⁻¹ (two right columns) at T = 260, 280 and 300 K are illustrated for simulations using the DLPNO_Myllys quantum chemistry data. The crosses mark conditions with both ∑J_2-comp and J_3-comp < 10⁻³ cm⁻³ s⁻¹.

The corresponding results for the RICC2 and DLPNO_Kubečka data sets are given in the ESI.† While the absolute formation rates predicted by these data are mostly very different (Fig. S1†), the qualitative biases in the simplifications are consistent (Fig. S2 and S3†). The absolute biases are affected by the predicted quantitative cluster stabilities: lower stabilities (DLPNO_Kubečka) mostly result in more bias and higher stabilities (RICC2) in less bias, with DLPNO_Kubečka generally giving highest biases up to factors of ∼40–60. As effective cluster stabilities depend on vapor concentrations and temperature (i.e. molecular growth vs. evaporation), the conditions with largest biases may be somewhat shifted in the vapor concentration space or T space for the different data sets.

Fig. 3 shows ∑J_2-comp/J_3-comp for the synthetic test chemistries H₂SO₄–base₁–base₂, where the two bases have exactly the same properties. That is, clusters with equal amounts of H₂SO₄ and base are assumed to be similar regardless of whether they contain base₁, base₂, or both, and the cases are thus correctly represented by lumping. Here the largest effects occur around the diagonal [base₁] : [base₂] = 1 : 1. For the weaker base represented by NH₃, the sum approach leads to underprediction by factors of ≲5 (260 K) and ≲10 (280 K, 300 K). For the stronger base DMA, underprediction (up to factors of ≲10) and possibly overprediction (up to a factor of ∼2) occur at low and high [base], respectively. Base concentrations with the largest overprediction, however, correspond to very polluted environments. These trends are understood through the non-linear behavior of J as a function of vapor concentration, here [base]: at low [base], increase in [base] causes a stronger increase in J than at high [base] where the system eventually becomes saturated with respect to [base]. That is, as the slope of J decreases with increasing [base], the sum rates ∑J_2-comp change from too low to too high. Under saturated conditions, the rate is double-counted by ∑J_2-comp at base₁ ≈ base₂, as the explicit formation rate remains the same upon doubling of base concentration. These patterns are similar for RICC2 (Fig. S4†) and DLPNO_Kubečka (not shown; very similar to DLPNO_Myllys), with some shifts in the vapor concentration space due to the differences in predicted stabilities. For example, as RICC2 predicts higher stabilities, the underprediction is often less pronounced compared to the other data sets and also slight overprediction may occur, since J is less sensitive to [base] and saturates at lower [base].

Fig. 3 Comparison of simulated particle formation rates assuming non-interacting base species (∑J_2-comp) or the fully interacting three-component system (J_3-comp) for similar bases, here using “ideal cases” of bases with identical thermochemical data (either NH₃ data or DMA data). Ratios of ∑J_2-comp/J_3-comp at different concentrations of the bases at two different [H₂SO₄] and CS of 10⁻³ s⁻¹ (two left columns) and 10⁻² s⁻¹ (two right columns) at T = 260, 280 and 300 K are illustrated for simulations using the DLPNO_Myllys quantum chemistry data: (a) for weakly clustering bases (NH₃); (b) for strongly clustering bases (DMA). Grey areas indicate that both ∑J_2-comp and J_3-comp are <10⁻¹⁰ cm⁻³ s⁻¹. The crosses mark conditions with both ∑J_2-comp and J_3-comp < 10⁻³ cm⁻³ s⁻¹.

Assessments of lumping of different bases, conducted with the DLPNO_Kubečka multi-base data, suggest that DMA and EDA are similar enough to be better described by lumping than summing. Fig. 4 shows J_lumped/J_3-comp, with J_lumped obtained assuming the properties of DMA and a total base concentration of [DMA] + [EDA] (see Fig. S6† for summing). Errors due to lumping are minor when [DMA] and [EDA] are of the same order, and become notable only at [DMA] : [EDA] ≪ 1 when EDA is the dominating species.

Fig. 4 Comparison of simulated particle formation rates using “lumped” base species, with EDA treated as DMA (J_lumped), or the fully interacting three-component H₂SO₄–DMA–EDA system (J_3-comp). Ratios J_lumped/J_3-comp at different concentrations of the bases at two different [H₂SO₄] and CS of 10⁻³ s⁻¹ (two left columns) and 10⁻² s⁻¹ (two right columns) at T = 260, 280 and 300 K are illustrated for simulations using the DLPNO_Kubečka quantum chemistry data. Grey areas indicate that both J_lumped and J_3-comp are <10⁻¹⁰ cm⁻³ s⁻¹. The crosses mark conditions with both J_lumped and J_3-comp < 10⁻³ cm⁻³ s⁻¹.

However, here the lumping of TMA with DMA/EDA gives errors of similar order or higher than the sum approach (Fig. S7†). This is likely due to TMA being less efficient in cluster growth, as its three methyl groups cause steric hindrance. MA lies between NH₃ and the other amines and thus cannot be lumped with any of these bases (e.g. Fig. S8†). Comparisons of lumping vs. summing for DLPNO_Kubečka are summarized in ESI Section 3†: for H₂SO₄–NH₃–amine chemistries, trends in the biases of the sum approach are similar to those for H₂SO₄–NH₃–DMA. For H₂SO₄–amine combinations, underprediction in summed J peaks when the amine concentrations are of the same order (except for MA for which biases peak at higher [MA]). For lumped J, patterns are less regular and wider ranges of conditions become biased—both low and high—when the species are not lumpable (e.g. Fig. S7†).

To find indicators for similarity, we compared the formation rates J_2-comp of the isolated systems (ESI Section 3†). The differences in J_2-comp generally follow the patterns of the error in the lumped rate J_lumped (e.g.Fig. 4); small differences (less than a factor of 10 between H₂SO₄–DMA and H₂SO₄–EDA) coincide with low errors in J_lumped. This suggests that the rates can be used as an indicator to assess similarity. In addition, a simple theoretical proxy based on acid·base heterodimer stability³⁶ is in line with the comparisons (ESI Section 3†), and can thus serve as an additional measure.

The quantitative biases may be affected by the presence of ionic species. Ion effects are assessed using the RICC2 data set, which includes also charged clusters, assuming an average boundary layer ion production rate of 3 cm⁻³ s⁻¹. For the H₂SO₄–NH₃–DMA case, ions do not have notable effects on the bias at CS = 10⁻³ s⁻¹ but significantly decrease the maximum biases at CS = 10⁻² s⁻¹, especially at T = 300 K (Fig. S2†). For the synthetic cases of identical bases, there are only minor effects for the strong base, but substantially less bias for the weaker base (Fig. S5†). This is due to J being less sensitive to vapor concentrations when the charge increases cluster stability. In general, ions are important for weaker clustering species and/or under conditions at which clusters are relatively less stable, such as low vapor concentrations and higher temperatures. Based on the RICC2 tests, ions can be expected to decrease the biases under such conditions.

To summarize, the commonly applied sum approach can be expected to give lower-limit assessments of the overall formation rate, here with the largest underprediction by factors of up to ∼10–100 (Table S2†) occurring at vapor level combinations at which all species are able to contribute (i.e. high and low concentrations for weak and strong clustering agents, respectively). The direction of the bias is consistent, as it follows from the omission of the total vapor concentration and similar behavior can thus be expected for other interacting chemistries. However, if model data and/or observations suggest very similar particle formation rates for stabilizing species with concentrations of the same magnitude, lumping is likely to involve less errors than summing of J.

3.2 Simplifications in cluster dynamics by assuming quasi-unary chemical systems

3.2.1 Applying effective rate constants obtained by the equilibrium assumption. To verify the applicability of the equilibrium approach in a case where the assumptions can be expected to be valid, we first performed test simulations for H₂SO₄–H₂O as water is an ideal example of such case. These tests show a perfect comparison (Fig. S9†), indicating that the implementation is robust and we can thus proceed to the H₂SO₄–base chemistries. Fig. 5 and 6 show the C_eq/C_expl ratios for each n-mer containing n H₂SO₄ molecules for NH₃ and DMA, respectively, for the RICC2 and DLPNO_Besel data. Here, C_eq is the concentration in the quasi-unary model system, and C_expl = ∑C_m is the total n-mer concentration summed over all base contents m in the explicit 2-compound simulation. The [base] ranges are here wider in order to recognize concentrations required for equilibration (in case it occurs).

Fig. 5 C _eq/C_expl ratios of n-mer concentrations ∑_m[(H₂SO₄)_n·(base)_m] obtained by assuming equilibration with respect to the base species (C_eq) and explicit multi-compound kinetics (C_expl) for H₂SO₄–NH₃ chemistry, modeled applying (a) the RICC2 and (b) the DLPNO_Besel quantum chemistry data set. The black diagonal lines correspond to [H₂SO₄] : [base] = 1 : 1, the rectangles give an approximative vapor concentration range for typical laboratory and ambient measurements, and the grey areas indicate n-mer concentrations below 10⁻¹⁰ cm⁻³. The crosses mark the conditions at which C_eq is insensitive to the quantitative evaporation rates (C_eq within a factor of two upon cluster stabilization by 2 kcal mol⁻¹).

Fig. 6 C _eq/C_expl ratios of n-mer concentrations ∑_m[(H₂SO₄)_n·(base)_m] obtained by assuming equilibration with respect to the base species (C_eq) and explicit multi-compound kinetics (C_expl) for H₂SO₄–DMA chemistry, modeled applying (a) the RICC2 and (b) the DLPNO_Myllys quantum chemistry data set. The black diagonal lines correspond to [H₂SO₄] : [base] = 1 : 1, and the rectangles give an approximative vapor concentration range for typical laboratory and ambient measurements. The crosses mark conditions at which C_eq is insensitive to the quantitative evaporation rates (C_eq within a factor of two upon cluster stabilization by 2 kcal mol⁻¹).

The main conclusions can be summarized as follows: first, all data sets show a consistent tendency towards overprediction of n-mer concentrations by the equilibrium assumption by factors of up to ∼10–10² and even ≳10³ especially at low [base]. The overprediction typically emerges and/or increases with increasing n and decreasing [base], although C_eq/C_expl may also exhibit maxima along the base axis, likely linked to shifts in the most abundant compositions (i.e. shifts in the growth paths; Fig. S10†). The positive bias can be understood through base dynamics: the equilibrium approach assumes that cluster formation and growth are not limited by the availability of base vapor, which is not the case at low [base], leading to overpredicted C_eq.

Second, the comparison is often seemingly better for the more strongly clustering species H₂SO₄–DMA. However, this is not due to the clustering kinetics (most importantly the evaporation rates; eqn (5)) being well described by the effective equilibrium rate constants. Instead, the evaporation rates of the H₂SO₄–DMA clusters are often so low compared to vapor collision rates that their exact values do not have major effects. This was tested by re-running the equilibrium simulations with all clusters stabilized by 2 kcal mol⁻¹, which decreases all evaporation rates by approximately two orders of magnitude. The conditions at which the n-mer concentrations C_eq,test of these test runs are within a factor of two from C_eq are marked in the figures with crosses, implying that the quantitative evaporation rates are not very important. Such cases include some of the low-temperature conditions for H₂SO₄–NH₃, and most conditions for H₂SO₄–DMA. This type of condition can be experimentally identified based on high cluster concentrations: for the crossed areas, C_expl ≳ 10⁴–10⁵ cm⁻³.

The figures can thus be interpreted as follows: under conditions with no bias and no crosses, the equilibrium assumption can be considered reasonable. Under conditions that have no bias but are marked with crosses, the quasi-unary approximation is reasonable but independent of the validity of the assumption, as well as of the exact rate constants. Under conditions with visible bias the clusters are not equilibrated with respect to base due to low [base] and/or evaporation effects, resulting in distorted predictions.

Also the weakly clustering H₂SO₄–NH₃ case may show a good performance for the equilibrium assumption, but this depends strongly on the conditions, the cluster size (n-mer), and data set (i.e. the exact evaporation rates). For such chemistries, the assumption is thus more uncertain and unreliable. Overall, the effects of temperature on the quantitative C_eq/C_expl vary somewhat with the species and data sets. Increasing temperature can facilitate equilibration by increasing evaporation rates and thus shortening evaporation time scales. On the other hand, increasing T may also introduce notable evaporation in a chemical system that has insignificant evaporation and thereby less bias at lower T. The quantitative C_eq/C_expl ratios are also affected by the scavenging sink: here, we apply a reference sink of CS = 10⁻³ s⁻¹, which is of the order of an average atmospheric sink or a wall loss sink in a clean chamber experiment.⁴⁹ Test simulations with 10⁻⁴ s⁻¹ and 10⁻² s⁻¹ show increasing overprediction with increasing CS (not shown). In general, scavenging effects become more significant at lower cluster concentrations which can occur, for example, at higher T, which may also contribute to the observed temperature trends.

The DLPNO_Kubečka data set that generally involves higher evaporation rates gives significantly more overprediction especially for the weaker clustering agents NH₃ and MA, but the quasi-unary approach becomes less reliable also for DMA, as well as for EDA (not shown). Within DLPNO_Kubečka, TMA at low T gives the least bias for the quasi-unary assumption (Fig. S11†). Simulations with H₂SO₄–GUA (DLPNO_Myllys) are in line with these results: the guanidine-enhanced system with extremely low evaporation rates shows a good comparison even at high T (Fig. S12†).

The bias and its size-dependence are directly linked to cluster survival probability assessments. Fig. 7 shows an example of a cluster survival parameter, here defined as the ratio SP = C_n/C_ref where “ref” is a reference size smaller than n, for H₂SO₄–NH₃ within DLPNO_Besel data. Here the survival parameter is overpredicted by the equilibrium approach, converging to the explicit results only at very high [NH₃]. Quantitative survival estimates obtained by such a quasi-unary model should thus be interpreted with caution especially if NH₃ is not in large excess.

Fig. 7 Cluster survival parameter, defined as SP = C_n/C_ref where the reference size is 3-mer, for the explicit and equilibrium simulations for H₂SO₄–NH₃ and DLPNO_Besel quantum chemistry data sets at 280 K. The vertical lines give an approximative vapor concentration range for typical laboratory and ambient measurements.

3.2.2 Approximating cluster properties by the most stable multi-compound composition. Despite the overprediction trend, the equilibrium approach cases suggest that the H₂SO₄–base chemistries can be reduced under specific conditions, and possibly even further simplified when the distributions with respect to base are dominated by a single composition (e.g. Fig. S10†). Fig. S13 and S14† show relative n-mer concentrations as in Fig. 5 and 6, now applying the assumption of the most stable composition (Section 2.3.2). However, the simplified approach leads to substantially worse overprediction in case of weaker cluster formation: for H₂SO₄–NH₃, cluster concentrations are overpredicted by factors of up to 10³ and above. The comparisons are better only at very high [base], mainly under conditions at which the simple quasi-unary results C_st are close to the equilibrium results C_eq and concurrently exact evaporation rates are unimportant (marked with plus signs and crosses, respectively). Similarly, C_st are close to C_eq for the low-evaporating case H₂SO₄–DMA, and positive bias mainly occurs at [H₂SO₄] : [DMA] ≳ 1 when clustering becomes base-limited, and at higher T.

Overall, while quasi-unary approximations are convenient for studying nanoparticle formation and growth dynamics, the results show that their applicability is limited to very strongly interacting, i.e. low-evaporating, species and/or low temperatures at excess base vapor. For weaker clustering agents such as H₂SO₄–NH₃, which can be expected to be important on the global scale, the formation process is affected by multi-compound kinetics. Representing effective cluster properties through the distribution of base contents results in less bias than using a single composition, but the concentrations are often still significantly overpredicted. Vice versa, fitting a quasi-unary model to reproduce measurements cannot be expected to give rate constants that correspond to actual effective cluster properties. While the comparisons improve with increasing [base], the base concentrations at which the approximation becomes valid are often beyond typical atmospheric ranges, corresponding to very polluted environments. Application of quasi-unary approaches for such clustering agents generally leads to overprediction of nanoparticle concentrations, growth rates and survival probabilities. Overpredicted concentrations may also result in, for instance, overpredicted contribution of cluster coagulation to nanoparticle growth.

4 Conclusions

The evaluations for H₂SO₄–base chemistries indicate that some typical approaches to simplify multi-component particle formation can often be applied without sacrificing much accuracy, while others are reasonable only for specific, limited conditions. The trends in the biases are consistent and similar for different quantum chemical data sets. We tested simplifications for different parameters that are central in particle formation modeling: formation rates that are used in atmospheric aerosol models, and molecular cluster properties that are applied to assess cluster growth and survival, as summarized in Table 3.

Table 3 Qualitative assessment of the applicability of different simplifications in multi-component particle formation modeling based on the representative H₂SO₄–base chemistries^a,b

Model parameter and simplification	Applicability for different types of compounds
Formation rate	Different clustering properties	Similar clustering properties
a A checkmark implies that the simplification can in principle be applied without major errors, and a cross indicates that the simplification leads to distorted results and should not be applied. The applicability can depend on ambient conditions, as indicated in footnote b. b *Underprediction is expected at [base1] ≫ [base2], where base1 and base2 are weaker and stronger stabilizers, respectively, and [base2] is at the ppt or sub-ppt level. **Lumping is reasonable especially at similar concentrations of the compounds; for summing, underprediction is expected when the base concentrations are of the same order: [base1] ∼ [base2]. ***Applicable at low temperatures (here mainly ≲280 K) and excess concentration of the implicitly treated compound.
Summing of rates
Lumping of species

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Cluster concentrations	Weakly clustering compounds	Strongly clustering compounds
Quasi-unary approximation by equilibrium assumption
Quasi-unary approximation by most stable composition

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The main results for formation rates J are as follows:

• For stabilizing compounds base₁ and base₂ with very different clustering efficiencies, the sum assumption J(H₂SO₄–base₁–base₂) ≈ J(H₂SO₄–base₁) + J(H₂SO₄–base₂) is biased towards underprediction for the combination of high [base₁] and low [base₂], where base₁ and base₂ are weaker and stronger stabilizers, respectively. The errors are minor especially for clean environments with low CS, but may increase to factors of up to ∼10–100 as CS increases to 10⁻³–10⁻² s⁻¹.

• As the clustering efficiencies of the species approach each other, the underprediction tendency shifts towards similar concentrations [base₁] ∼ [base₂].

• For very similar compounds, J is reasonably approximated by lumping the species through an effective vapor concentration [base] = [base₁] + [base₂]. The sum approach can result in errors of factors of up to ∼10–100 when [base₁] and [base₂] are of the same order. Under- and overprediction by the sum approach typically occur at lower and higher [base], respectively; the overprediction is minor and mainly occurs for strongly clustering species when J is saturated with respect to base.

• Similarity can be assessed based on formation rates, and additionally by a theoretical proxy involving the free energies of acid·base heterodimers. Minor differences in these indicators, here approximately by factors of ≲10, suggest similarity.

• A recent multi-base data set suggests that dimethylamine and ethylenediamine are similar enough to be lumped. The properties of trimethylamine are somewhat different compared to those of these two other amines, and lumping does not generally give better results than the sum approach.

• The presence of ions is expected to decrease the biases in the simplified formation rates. Charge enhances cluster binding, which reduces the differences in the relative stabilization efficiencies of different species.

Reducing H₂SO₄–base chemistries to quasi-unary H₂SO₄ systems for modeling molecular cluster concentrations is, in contrast, much more uncertain:

• The quasi-unary approaches are mainly applicable for strong stabilizing species with low-evaporating clusters at vapor concentrations that are high enough to not limit the cluster formation. Such a case is, for example, an efficiently clustering H₂SO₄–amine chemistry at [amine] : [H₂SO₄] > 1.

• Especially for weaker clustering species such as H₂SO₄–NH₃ that are common particle formation agents on the global scale, and/or for warm conditions, cluster concentrations are overpredicted by up to 1–2 orders of magnitude, or even more, in the quasi-unary frameworks.

• The biases in cluster concentrations affect particle survival probability assessments, which can also be biased high, as the overprediction often emerges and/or increases with cluster size.

Simplifications in multi-compound chemistries are highly desirable for convenient representation, and the tests suggest that they can be possible especially for formation rates. The present results provide assessments of how and when to simplify and of the possible biases, thereby contributing to improved implementation of input data and understanding of model uncertainties.

Author contributions

Conceptualization, software (kinetic modeling), methodology: TO; formal analysis, investigation, visualization: RB and TO; resources (quantum chemical data and analysis): JK, NM, and JE; writing – original draft: TO; writing – review & editing: all authors.

Conflicts of interest

There are no conflicts of interest to declare.

Acknowledgements

T. O. and R. B. acknowledge the Swedish Research Council VR (grant no. 2019-04853) and the Swedish Research Council for Sustainable Development FORMAS (grant no. 2019-01433) for financial support. J. K. and J. E. thank the Independent Research Fund Denmark (grant no. 9064-00001B). N. M. thanks the Academy of Finland for funding (grant no. 347775). We thank the CSC-IT Center for Science in Espoo, Finland, for providing computational resources.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d2ea00174h

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