The dependence of new particle formation rates on the interaction between cluster growth, evaporation, and condensation sink

Abstract

New particle formation (NPF) is one of the major contributors to atmospheric aerosol number concentrations. The initial step of NPF includes the formation and growth of small clusters, their evaporation and loss to pre-existing particles (characterized by the condensation sink, CS). In the polluted atmospheric boundary layer, the high environmental CS suppresses NPF and it can work synergistically with evaporation to further reduce the NPF rates. In this study, to quantitatively include CS into NPF analysis, we make simplifications to the cluster balance equations and develop approximate equations for the NPF rates in the presence of pre-existing particles, which are applicable to nucleation mechanisms that can be represented by a nonbranched nucleation pathway. The developed equations show that the proportion of clusters that finally lead to new particle formation is given by the cluster-specific ratio of . As a result, the cumulative product of this ratio for all clusters in the nucleation pathway determines the NPF rates. By comparing with benchmark cluster dynamics simulations of sulfuric acid-dimethylamine and sulfuric acid-ammonia nucleation systems, the developed equations were confirmed to give good estimates of the NPF rates and approximately capture the dependency of NPF rates on CS and nucleating vapor concentrations. The CS dependency predicted by the developed equations shows larger deviations from the simulations when the cluster evaporation rates are high, i.e., when the underlying assumptions of the equations are not satisfied. The equations were also found to be in good agreement with atmospheric NPF rates measured in long-term field observations in urban Beijing.

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Environmental significance

Atmospheric new particle formation (NPF) affects the climate by influencing cloud condensation nuclei number concentrations and is associated with haze formation, which negatively impacts the environment and human health. NPF is initiated by the formation of clusters composed of precursor vapor molecules, such as sulfuric acid, amines, ammonia, and organics. On their way to become nucleated particles, these clusters can grow by molecular addition, evaporate, or be scavenged by pre-existing particles in the atmosphere. In this work we show how these processes collectively determine the NPF rates by deriving analytical rate equations. The equations derived share a common physical interpretation, i.e., the proportion of clusters to become aerosol particles depends on the ratio of the cluster growth rate to the sum of the growth rate, the evaporation rate and loss rate to pre-existing particles. The equations are verified by comparison with benchmark cluster dynamics simulations and are found to predict reasonably accurate NPF rates for a long-term campaign in Beijing.

1 Introduction

New Particle Formation (NPF) and the subsequent growth of nucleated particles are globally a major contributor to cloud condensation nuclei and are associated with heavy particulate pollution episodes in megacities.^1,2 The occurrence of NPF is governed by several factors, e.g., the concentration of the nucleating vapors, the ambient temperature, and the background particle concentrations. Among these factors, the loss of nucleating vapors/clusters to pre-existing particles, often characterized by the condensation sink (CS), plays a critical role in polluted regions since a high CS strongly suppresses NPF by decreasing the survival probability of the growing clusters.^3–7 Therefore, CS is an essential part of NPF models formulated to explain field observations.^8–10 Recent studies also focused on the accurate calculation of CS. Tuovinen et al.¹¹ studied the dependence of the condensation sink on condensing cluster (vapor) properties and discussed the possibility that CS is overestimated.

In the presence of pre-existing particles, NPF rates depend on the interaction between cluster growth, evaporation, and CS. With CS added to the picture, analytical tools of nucleation, which were developed only considering cluster growth and evaporation, require re-evaluation. These tools include the classical nucleation theory and the nucleation theorems.^12,13 As an extension to the classical nucleation theory, McGraw and Marlow¹⁴ derived closed-form expressions for homogeneous nucleation rates in the presence of pre-existing particles. Malila et al.¹⁵ later revised the first nucleation theorem and developed sum rules, which relate the size-dependent nucleation rates, cluster loss and the CS-free nucleation rates. These studies provide valuable insights into how CS alters the nucleation process but are pivoted towards the modification of the classical theories. This leads to relatively complex expressions for the nucleation rates whose application to atmospheric NPF is not straightforward.

The objective of this work is to better understand the interplay between cluster growth, evaporation, and CS by deriving approximate equations for size dependent NPF rates with clear physical interpretations. This is made possible by simplifications of the cluster population balance equations, which are fair approximations in typical ranges of nucleating vapor concentrations, cluster evaporation rates and CS in the atmospheric boundary layer. Based on these equations, we discuss the competition between the cluster growth and sink, the synergy of evaporation and CS to suppress NPF, and the power dependencies of the NPF rates on CS and nucleating vapor concentrations. In addition, to verify and elaborate the derived equations, we compare the theoretical equations with benchmark cluster dynamics simulations of the sulfuric acid-dimethylamine (SA-DMA) and the sulfuric acid-ammonia (SA-AM) nucleation systems. The former has been identified to be a dominant nucleation mechanism in polluted megacities in China,^8,16 while the latter is a representative nucleation system with considerable cluster evaporation. Finally, the developed equations were found to give good estimates of NPF rates measured in Beijing in long-term field observations from 23 January 2018 to 16 January 2019.

2 Theoretical analysis

We consider NPF along a nonbranched nucleation pathway as shown in Fig. 1, which allows different vapor molecules to add to or dissociate from clusters. This schematic is an extension of the homo-molecular nucleation scenario and incorporates important multicomponent nucleation systems found in the atmospheric boundary layer. For instance, at high base concentrations, the addition of the acid molecules is the rate-limiting step for acid–base nucleation, and hence acid–base nucleation has been proposed to follow a mechanism similar to that in Fig. 1.⁹ The coagulation of clusters is not included, and we shall assess the role of coagulation as we continue our analysis with examples. In terms of the ‘length’ of the nucleation pathway, we consider pathways involving a few steps, as numerous studies have suggested that the first stable clusters in NPF events only contain a few molecules.^17–19 To estimate the NPF rates at larger particle sizes (e.g., above 2 nm), continuous approaches^20,21 that consider the effect of CS can be coupled with the current method. The term ‘coagulation sink (CoagS)’ is often used to characterize the loss of clusters and particles to pre-existing particles.^3,22 Here we use CS to represent both CS and CoagS for brevity, since the calculation methods of CS and CoagS are essentially identical.

Fig. 1 Illustration of the nucleation scenario considered in this work. Spheres of different colors indicate the same or different vapor molecules.

2.1 CS and the NPF rates in the absence of evaporation

We start by considering a simplified case in which all clusters are non-evaporative. Assuming that the cluster distribution has reached a steady state, the population balance equation for the ith cluster (i.e., cluster i) along the nucleation pathway is

k_i−1n_i−1 = k_in_i + CS_in_i (i ≥ 2), (1)

where n_i is the concentration of cluster i, k_i is the cluster growth rate (which is equal to the product of the collision rate coefficient and the vapor concentration) and CS_i is the condensation sink for cluster i. Rearranging eqn (1) leads to

Using the above recursive equation, the relationship between n_m−1 and n₁ is given by

The NPF rate J_m is defined as the formation rate of cluster m,

Note that throughout this work the subscript of J refers to the cluster size rather than the particle diameter, which is often the case in the NPF literature. Eqn (3) and (4) show that the steady-state cluster concentration and the NPF rate critically depend on (or CS_i/k_i) since the terms containing this ratio are multiplicative. For highly polluted environments satisfying the condition up to cluster m − 1, eqn (4) reduces to

The last equality in eqn (5) is valid for a homo-molecular system and β_i is the collision rate coefficient between cluster i and the monomer. According to eqn (5), the power dependency of J on monomer concentration in an evaporation-free, homo-molecular system strongly depends on m, i.e., the cluster size at which NPF rates are defined/measured. A larger m means that the clusters have to travel a longer pathway to be counted as nucleated particles, while being scavenged by pre-existing particles, leading to a stronger dependence of J_m on n₁.

In the above derivation, cluster–cluster coagulation is not included but can potentially contribute a significant portion of the new particles. To assess if coagulation strongly affects the accuracy of eqn (4), Fig. 2 compares J₄ and J₆ calculated by using eqn (4) (J_{eqn (4)}) and by cluster dynamics simulations, which include both molecular addition and cluster–cluster coagulation (J_sim, see the ESI† for more details). Fig. 2 shows that J_{eqn (4)} differs from J_sim by less than a factor of 1.5 for J₄ and by less than a factor of 2 for J₆ under the conditions n₁ = 10⁶ to 5 × 10⁷ cm⁻³ and CS = 0.001–0.1 s⁻¹. It is noteworthy that although coagulation adds extra channels for cluster formation, it does not necessarily lead to higher NPF rates than eqn (4) because cluster coagulation competes with cluster growth by molecular additions: if clusters are consumed by coagulation to form new particles, there are correspondingly fewer clusters to grow by molecular additions. Additionally, as shown by Fig. S1 in the ESI,† particle formation by coagulation is more suppressed than particle formation by molecular addition as CS increases.

Fig. 2 J ₄ (A) and J₆ (B) as functions of the nucleating vapor concentration for various CS values in a homo-molecular system. Note that here ‘4’ and ‘6’ in the subscript of J refer to the number of molecules in the cluster. The NPF rates are calculated by using both eqn (4) (i.e., J_{eqn (4)}) and cluster dynamics simulations (J_sim). The power dependencies of J₄ and J₆ on n₁, which are extracted through linear fits on the log–log scale, are shown near the respective curves.

Fig. 2 also shows the power dependencies of J₄ and J₆ on n₁, which is extracted through linear fits on the log–log scale. As suggested by eqn (5), the power dependency of J₄ on n₁ approaches 4 and the power dependency of J₆ on n₁ approaches 6 as CS increases. We note that the fitted power dependencies of J_{eqn (4)} and J_sim on n₁ differ only slightly (shown as numbers near the respective fitting lines), which means that eqn (4) can give the correct power dependencies even if cluster–cluster coagulation is not included in its derivation.

In the interpretation of NPF data, the collision-controlled limit (J ∝ n₁²) is often used as a reference to determine if significant cluster evaporation occurs along the nucleation pathway.^23–25 The above analysis shows that a high CS drives the system away from this limit, with the extent of deviation further influenced by the particle size at which NPF rates are measured. As a result, the measured NPF rates and the collision-controlled limit are not expected to agree in high CS regions even in the absence of cluster evaporation. This means that the deviation of NPF rates from the collision-controlled limit does not show the nature of the rate-limiting factors, which could be CS or cluster evaporation or both.⁸ Only by comparing NPF data measured within a narrow range of CS with one of the theoretical curves in Fig. 2 can the influence of cluster evaporation be extracted.

2.2 CS, cluster evaporation and NPF rates

A more general case in which clusters do evaporate is considered. At the steady state, the population balance equation for cluster i is

k_in_i + CS_in_i + E_in_i = k_i−1n_i−1 + E_i+1n_i+1 (i ≥ 2). (6)

where E_i is the evaporation rate for cluster i. Assuming that clusters larger than cluster m − 1 do not evaporate, then for cluster m − 1,

k_m−1n_m−1 + CS_m−1n_m−1 + E_m−1n_m−1 = k_m−2n_m−2. (7)

With eqn (6) and (7), the relationship between n_m−1 and n₁ can be derived. The NPF rate, defined as the formation rate of cluster m, is then given by J_m = n_m−1k_m−1.

We derived the expressions for J₄ and J₅ as representative examples. In the derivation of J₄, the tetramer is assumed to be non-evaporative; in the derivation of J₅, the tetramer is evaporative but the pentamer is assumed to be non-evaporative. (In real nucleation systems, the non-evaporative cluster size depends on the cluster stability under ambient conditions, which needs to be retrieved from measurements^9,19 or predicted by, e.g., quantum chemistry) Here we discuss J₄ and the expression for J₅ is found in the ESI.† Using eqn (6) and (7), it can be shown that the relationship between n₃ and n₁ is

Therefore,

The denominator of eqn (9) is the summation of three terms: the first term shows the competition between CS and cluster growth, the second term indicates the competition between evaporation and cluster growth, and the third term embodies the synergistic effect of cluster evaporation and CS to quench NPF. It is due to the third term that CS has a stronger quenching effect on NPF in the presence of cluster evaporation. Assuming CS₂ = CS₃ = CS (CS inversely scales with the square root of cluster mass, and hence CS_1–3 are expected to differ by less than 45% for a homo-molecular system) further reduces the third term to , which clearly indicates that the synergistic effect of CS and evaporation depends on if there are strongly evaporative clusters along the nucleation pathway. The simultaneous effects of evaporation and CS can be qualitatively understood by considering the time that a nucleated particle has to ‘stay’ as a certain cluster: if a cluster is strongly evaporative, it takes a longer time for the nucleated particles to grow beyond this cluster and are more likely to be scavenged by the pre-existing particles.

Eqn (6) can be simplified in the limit of E_i+1n_i+1 ≪ k_i−1n_i−1. This limit is satisfied if either of the following conditions is met. First, if cluster i + 1 is exceedingly stable (E_i+1 ∼ 0), E_i+1n_i+1 approaches zero and can be neglected. Second, if cluster i + 1 is comparatively much more stable than cluster i (E_i ≫ E_i+1), and n_i+1 is not much larger than n_i, we have k_i−1n_i−1 > E_in_i ≫ E_i+1n_i+1 (k_i−1n_i−1 − E_in_i is the net flux from cluster i − 1 to cluster i, which is positive when NPF occurs). With E_i+1n_i+1 ≪ k_i−1n_i−1, eqn (6) reduces to

k_in_i + CS_in_i + E_in_i = k_i−1n_i−1. (10)

The relationship between n_m−1 and n₁ can be obtained by solving eqn (10) and (7):

The NPF rate is

Eqn (12) provides an easy-to-implement formula to estimate the NPF rate if the thermodynamic data of the clusters are available (hence E can be estimated). To test the accuracy of eqn (12), we compared the exact solutions for J₄ (i.e., eqn (9)) and J₅ (see the ESI†) with values computed with eqn (12) for a homo-molecular system. These results are shown in Fig. S2 and S3 in the ESI.† For all the conditions we have tested, the maximum deviation of eqn (12) from the exact solutions is less than 25% for J₄ and less than 45% for J₅, which is small compared to the typical uncertainty of NPF rate measurements.²⁶ Despite the good agreement, it should be noted that in nucleation scenarios with long nucleation pathways and high backward fluxes due to cluster evaporation (i.e., the underlying assumptions of eqn (12) are violated), the application of eqn (12) could introduce more significant biases. However, as we show later in Section 3, eqn (12) can predict fairly accurate nucleation rates for atmospherically relevant nucleation systems.

Eqn (12) shows that the NPF rate depends critically on the cumulative product of the terms for different-size clusters. On the one hand, this dependency indicates that CS and E work synergistically to reduce NPF rates through the multiplication of their summation; on the other hand, it gives a clear physical interpretation of the cluster formation process. For each cluster along the nucleation pathway, the cluster can grow, decay to the previous cluster, or lose to pre-existing particles. Eqn (12) indicates that the proportion of clusters that finally contribute to nucleation is (to a good approximation) simply the ratio of the growth flux to the sum of all three fluxes. Eqn (12) also has implications for the dependency of NPF rates on CS. For instance, J₄ is proportional to according to eqn (12). If CS ≫ E + k for both clusters 2 and 3, then . On the other hand, if E + k ≫ CS for clusters 2 and 3, J₄ barely scales with CS. For intermediate conditions, the dependency of J₄ on CS is between non-dependent and inversely quadratic.

3 Comparison with cluster dynamics simulations

In this section, we compare variants of eqn (12) to benchmark cluster dynamics simulations. Specifically, we examine sulfuric acid-dimethylamine (SA-DMA) and sulfuric acid-ammonia (SA-AM) binary nucleation. The former has been confirmed to be a dominant NPF mechanism in polluted megacities with a high background CS in China,^8,16 which is hence highly relevant to NPF under high CS conditions. The latter is a binary system with high cluster evaporation rates, which tests the capability of eqn (12) when its underlying assumptions are violated to some extent.

3.1 Simulation setup

Cluster growth, evaporation, coagulation, and the condensation sink are included in the simulation. Cluster population balance equations are numerically solved with a home-built MATLAB code, with cluster evaporation rates computed using the Gibbs free energy of cluster formation and collision coefficients with the method presented in McGrath et al.²⁷ The Gibbs free energies of formation for the SA-DMA clusters and the SA-AM clusters are available from several sources.^28–30 Here we use the more recent data published by Li et al.³⁰ computed at the RI-CC2/aug-cc-pV(T+d)Z//M06-2X/6-311++G(3df,3pd) level of theory. We note that the RI-CC2/aug-cc-pV(T+d)Z level of theory tends to overpredict the cluster binding energy, which in turn leads to underpredicted evaporation rates.³¹ However, NPF rate calculations based on this level of theory have shown good agreement with measurements and observations.^32,33 Additionally, in the following we conduct simulations at different temperatures (see below) so that the simulation-theory comparison includes a representative range of cluster evaporation rates. The cluster specific CS is calculated by integrating the cluster collision rates with an artificial background particle distribution and variation of CS is achieved by uniformly scaling the distribution by a factor. A uniform collision enhancement factor of 2.3 is assigned to collisions between molecules and clusters.⁸ All the NPF rates reported below are calculated after the cluster distribution reaches a steady state in the simulation.

For the SA-DMA nucleation, simulations were performed at 298 K and 280 K at fixed nucleating vapor concentrations as listed in Table 1A. Two temperatures were chosen since at the lower temperature the cluster evaporation rate is low and NPF is close to the non-evaporative limit, while at the higher temperature cluster evaporation markedly slows NPF. Note that [SA]_t in Table 1A is the combined concentration of SA₁ and SA₁DMA₁. Current instrumentations deployed in field measurements cannot distinguish SA₁DMA₁ from SA₁, and hence field measurements report [SA]_t rather than the concentration of SA₁.¹⁸ We fix [SA]_t in the simulation to match this constraint in field measurements. The simulation domain contains clusters of SA_xDMA_y with y ≤ x ≤ 3. Clusters containing 4 or more sulfuric acid molecules are treated as particles and leave the simulation domain. The cut-off of the simulation domain approximately corresponds to the instrument detection limit of DEG-SMPS.^31–33 This simulation setting implies that clusters containing four or more SA molecules are stable. Since our goal for this section is to compare the theoretical equations to simulations, we ignore the possibility that SA_≥4DMA_y can be evaporative here.

Table 1 Cluster dynamics simulation conditions for Section 3. In (A), [SA]_t is the combined concentration of SA₁ and SA₁DMA₁, and [DMA] is the concentration of DMA molecules. E₁ is the evaporation rate of SA₁DMA₁. In (B), [SA] is the concentration of SA monomers and [AM] is the concentration of ammonia molecules. E₁ is the evaporation rate of SA₁AM₁; is the evaporation rate of SA₂. The evaporation rates of other clusters in the simulation are computed using the standard Gibbs free energy of cluster formation and collision rate coefficients (not listed in this table). The CS value for all simulation conditions span from 0 to 0.06 s⁻¹

(A)
	Case no.	280 K, E1 = 0.032 s^−1		298 K, E1 = 0.48 s^−1
		[SA]t (cm^−3)	[DMA] (cm^−3)	[SA]t (cm^−3)	[DMA] (cm^−3)
SA + DMA	1	2.5 × 10^6	2.5 × 10^7	2.5 × 10^6	2.5 × 10^7
	2	2.5 × 10^6	5.0 × 10^7	2.5 × 10^6	5.0 × 10^7
	3	5.0 × 10^6	5.0 × 10^7	5.0 × 10^6	5.0 × 10^7
	4	5.0 × 10^6	1.0 × 10^8	5.0 × 10^6	1.0 × 10^8
	5	[SA]t = 1.0 × 10^6–5.0 × 10^7 cm^−3, [DMA] = 2.5 ×10^7, 5.0 × 10^7, and 1.0 × 10^8 cm^−3

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(B)
	Case no.	270 K, E1 = 6.77 × 10^3 s^−1,		280 K, E1 = 2.0 × 10^4 s^−1,		298 K, E1 = 1.1 × 10^5 s^−1,
		[SA] (cm^−3)	[AM] (cm^−3)	[SA] (cm^−3)	[AM] (cm^−3)	[SA] (cm^−3)	[AM] (cm^−3)
SA + AM	1	1.0 × 10^6	3.0 × 10^9	2.5 × 10^6	1.0 × 10^10	1.0 × 10^7	1.0 × 10^11
	2	2.0 × 10^6	5.0 × 10^9	5.0 × 10^6	1.0 × 10^10	2.0 × 10^7	1.0 × 10^11
	3	2.5 × 10^6	1.0 × 10^10	5.0 × 10^6	2.5 × 10^10	2.0 × 10^7	2.0 × 10^11
	4	5.0 × 10^6	1.5 × 10^10	1.0 × 10^7	5.0 × 10^10	4.0 × 10^7	2.0 × 10^11

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For SA-AM nucleation, simulations were performed at 270 K, 280 K and 298 K at fixed nucleating vapor concentrations (Table 1B). Compared to the simulation of the SA-DMA nucleation, a temperature of 270 K is added to bring the highly evaporative SA-AM system closer to the non-evaporative limit. The concentration of SA₁ (rather than SA₁ + SA₁AM₁) is fixed in the simulation because [SA₁AM₁] is negligible compared to [SA₁] (SA₁AM₁ is highly evaporative as indicated by the E₁ values in Table 1B). The simulation domain contains clusters of SA_xAM_y with y ≤ x ≤ 4 as well as SA₁AM₂, SA₂AM₃ and SA₃AM₄. Clusters containing 5 or more sulfuric acid molecules are treated as particles and leave the simulation domain.

3.2 SA-DMA nucleation

To apply the analytical equations derived in Section 2 to SA-DMA nucleation, we specify a nonbranched nucleation pathway as shown in Fig. 3, similar to the mechanism proposed by Jen et al.⁹ Note that k₀ in Fig. 3 is calculated based on the concentration of DMA (i.e., k₀ = β_SA-DMA[DMA]), while k_1-3 are calculated based on [SA]_t. Using eqn (12), the NPF rate is given bywhere n_SA1 is the concentration of SA₁ and the meanings of the other variables are given in Fig. 3. At the steady state, [SA₁] and [SA]_t are related by (see the ESI† for its derivation),where n_t represents [SA]_t. Substituting eqn (14) into eqn (13) gives

Fig. 3 The nucleation pathway for the SA-DMA system. β_SA−DMA is the collision rate coefficient between SA and DMA molecules. β₁–β₃ are the collision rate coefficients between SA₁DMA₁, SA₂DMA₂, SA₃DMA₃ and SA₁DMA₁, respectively; k₁–k₄ are the cluster growth rates; CS₁–CS₃ are the loss rates to pre-existing particles; E₁–E₃ are the cluster evaporation rates.

Eqn (15) relates the NPF rate J to [SA]_t, which is held constant in the simulation. In the implementation of eqn (15) (but not in the simulation), we further assume that the clusters SA₂DMA_≤2 and SA₃DMA_≤3 are non-evaporative, i.e., E₂ = E₃ = 0. Based on the cluster free energies from Li et al.,³⁰ the evaporation rate of SA from SA₂DMA₁ is negligible, but the evaporation rate of SA from SA₃DMA₂ is not (see Table S1 in the ESI†). Therefore, setting E₂ = 0 should affect the calculation negligibly but setting E₃ = 0 should introduce some errors. However, as shown below, even with this simplification, eqn (15) can give fairly accurate predictions of the NPF rates.

The comparison between the simulation and eqn (15) is summarized in Fig. 4. Fig. 4A and B compare the simulated NPF rates (J_sim) and the rates calculated by using eqn (15) (J_{eqn (15)}) for CS = 0–0.06 s⁻¹. Fig. 4A and B show that as CS increases, the NPF rates decrease by 1–2 orders of magnitude, with a larger decrease for lower concentrations of the nucleating vapor. J_{eqn (15)} gives a very good estimation of J_sim at 280 K but slightly overestimates the NPF rates at 298 K by less than a factor of 3. The deviation of J_{eqn (15)} from J_sim could partially be influenced by the neglection of particle coagulation. However, as shown in Fig. S4 in the ESI,† particle formation by coagulation is less than 36% for all cases examined and is less than 25% for case 1 at 298 K, in which the maximum deviation occurs. Compared to coagulation, the assumption E₃ = 0 impacts the accuracy of J_{eqn (15)} more strongly. As mentioned above, the evaporation rate of SA₃DMA₂ is non-negligible at 298 K and can lead to the decay of this cluster if the DMA concentration is not high enough to instantly combine with it to form SA₃DMA₃. This causes J_{eqn (15)} to deviate further away from J_sim under low [DMA] conditions (case 1) than high [DMA] conditions (case 4) at 298 K.

Fig. 4 (A) and (B) Comparison of the simulated NPF rates J_sim and the theoretical rates J₍₁₅₎ (calculated by using eqn (15)) at 280 K and 298 K. Red, purple, blue and green curves/markers correspond to cases 1–4 listed in Table 1, respectively. (C) and (D) The dependency of J_sim and J_{eqn (15)} on CS for cases 1–4 at 280 K and 298 K. (E) J_sim of the SA-DMA system at 280 K at six CS values. Dashed black lines are linear fits to the curves on the log–log scale and the numbers above the curves are the power dependency of J_sim on [SA]_t. (F) Comparison of the power dependency of the NPF rates on [SA]_t. Fitted p is obtained by fitting the simulated NPF rates as shown in (E), while p₍₁₅₎ is obtained by derivatizing log(J_{eqn (15)}) with respect to log([SA]_t) at [SA]_t = 4.5 × 10⁶ cm⁻³.

Fig. 4C and D show the power dependency of J_{eqn (15)} and J_sim on CS for simulations at 280 K and 298 K, respectively. The power dependency is calculated by derivatizing the logarithm of J with respect to the logarithm of CS. Both Fig. 4C and D show that J_{eqn (15)} approximately captures the dependency of NPF rates on CS, with deviations from the simulation by less than 0.4. The NPF rates have a stronger dependency on CS at lower vapor concentrations and higher CS values. This trend can be explained by examining eqn (15), which shows that the NPF rate is proportional to . Both the decrease of vapor concentration (which leads to lower k's) and the increase of CS make the CS term more dominant in the denominator, and hence its variation more strongly affects the NPF rates. We note that at 280 K, CS₁ is comparable to E₁; at 298 K CS₁ is considerably smaller than E₁ (see Table 1A). This makes the power dependency on CS higher at 280 K than at 298 K. The variation of the CS power dependency indicates that there is no simple, universal way to scale the NPF rates by CS alone to facilitate the intercomparison of NPF rates measured in a wide range of CS, because the dependency relies on the relative magnitude of CS to cluster evaporation and growth rates. However, as suggested by Fig. 4C and D, the NPF rates in general become less dependent on CS as the dominant rate limiting factor transitions from CS to cluster evaporation rates.

The power dependency of J_{eqn (15)} and J_sim on [SA]_t is shown in Fig. 4E and F. To obtain this dependency, we simulated the steady state NPF rates under conditions of case 5 in Table 1A, spanning the typical ranges of SA and DMA concentrations during NPF events. Six CS values ranging from 0.001 s⁻¹ to 0.06 s⁻¹, which cover the CS values from clean atmospheres to severely polluted environments, were used in the simulations. The variation of J_sim as a function of [SA]_t at a DMA concentration of 5 × 10⁷ cm⁻³ at 280 K is shown in Fig. 4E. Linear fits are applied to the J_simvs. [SA]_t data on the log–log plot (black dashed lines in Fig. 4E), with their slopes representing the power dependency of J on [SA]_t (shown as numbers above respective lines). As CS increases, the power dependency increases and approaches a limiting value of 4. Fig. 4F shows a comparison of the power dependency of J_{eqn (15)} and J_sim on [SA]_t for all conditions examined in case 5. The x-axis is obtained by linear fits of log(J_sim)–log([SA]_t) as is done in Fig. 4E, while the y-axis is calculated by derivatizing log(J_{eqn (15)}) with respect to log([SA]_t) at [SA]_t = 4.5 × 10⁶ cm⁻³, which is the mid-point of [SA]_t on the log scale. Fig. 4F clearly shows that eqn (15) accurately captures the dependency of NPF rates on [SA]_t at both temperatures.

3.3 SA-AM nucleation

Unlike the SA-DMA nucleation in which the evaporation of many clusters is negligible, in the SA-AM nucleation the cluster evaporation rates can be significantly higher than their growth rates or CS. Here we model the SA-AM nucleation process with the schematic shown in Fig. 5. We consider two pathways that lead to the formation of SA₂AM₁ clusters, i.e., SA → SA₁AM₁ → SA₂AM₁ and SA → SA₂ → SA₂AM₁. Since SA₁AM₁ is highly unstable, the latter route can contribute significantly to NPF. We further simplify the evaporation kinetics of SA from SA_xAM_y clusters (x = 2–4, 1 ≤ y ≤ x) by assuming that clusters containing the same SA molecules but different AM molecules are at an equilibrium concentration (e.g., clusters in the dashed purple rectangle), recognizing that under typical atmospheric conditions the AM concentration is orders of magnitude higher than SA. We again apply eqn (12) and the NPF rate for the SA-AM system is expressed aswhere Ē_i is given by

Ē₂ = f₂₁(E_21,SA + E_21,AM) + f₂₂E_22,SA, (17a)

and

Fig. 5 The nucleation pathway for the SA-AM system. β is the collision rate coefficient: β₂–β₄ are the collision rate coefficients between SA₂AM₂, SA₃AM₃, SA₄AM₄ and SA₁, respectively, while the subscripts of other collision rate coefficients indicate the colliding molecules/clusters. k, CS and E represent the cluster growth rate, the condensation sink and the evaporation rate, respectively. The evaporation rates with a bar are calculated with eqn (17).

In eqn (16) and (17), E_ij,X is the evaporation rate of an X molecule from the cluster SA_iAM_j, f_ij is the fraction of SA_iAM_j of all clusters containing i SA molecules, and the meaning of other variables is given in Fig. 5. More details on the calculation of f_ij can be found in Section S6 in the ESI.†

Fig. 6A–C compare the simulated NPF rates (J_sim) and the rates calculated by using eqn (16) (J_{eqn (16)}) for CS = 0–0.06 s⁻¹ at 270 K, 280 K and 290 K, respectively. The vapor concentrations at each temperature were chosen so that J_sim at CS = 0 approximately lies within the atmospherically relevant range of 0.1 to 1 × 10³ cm⁻³ s⁻¹, and thus the vapor concentrations in cases 1–4 are different at different temperatures (Table 1B). Fig. 6A–C show that in all simulation cases, J_sim and J_{eqn (16)} are close in value within a difference less than a factor of 3. The agreement of J_sim and J_{eqn (16)} is better at higher vapor concentrations (i.e., cases 3 and 4 at 270 K and 280 K; cases 1–4 at 298 K), indicating that eqn (16) better predicts the NPF rates at higher vapor concentrations under the selected NPF conditions. Further examination of the NPF rate variation with respect to CS shows that J_sim and J_{eqn (16)} respond similarly to CS at higher NPF rates and lower temperatures (i.e., case 4 at 270 K and 280 K), while in other simulation cases J_{eqn (16)} is less sensitive to CS than J_sim, which shows a steeper downward trend as CS increases.

Fig. 6 (A–C) Comparison of the simulated NPF rates J_sim and the theoretical rates J_{eqn (16)} at 270 K, 280 K and 298 K. Shaded areas are plotted to aid the visualization of the difference between J_sim and J_{eqn (16)}. (D–F) The power dependency of J_sim and J_{eqn (16)} on CS at 270 K, 280 K and 298 K. The simulation conditions (Table 1B) are color-coded by case numbers (shown in panel A). Note that the vapor concentrations in cases 1–4 are different at different temperatures (Table 1B).

To better understand the relationship between the NPF rate and CS, we show the power dependency of J_sim and J_{eqn (16)} on CS in Fig. 6D–F. The power dependency was calculated by derivatizing the logarithm of the NPF rates with respect to the logarithm of CS. At 270 K and 280 K, the power dependency of J_{eqn (16)} follows J_sim closely, with the power dependency at low NPF rates (case 1, red curves) higher than the power dependency at high NPF rates (case 4, green curves). This trend can be explained by examining eqn (16): lower NPF rates correspond to lower cluster growth rates in our simulations, which means that the relative magnitude of CS in the denominator of eqn (16) is large and influences the NPF rates strongly. Fig. 6D and E also show that the power dependency does not exceed 3 at 270 K and 280 K, although there are four terms containing CS in the denominator of eqn (16). This is caused by the very large value of E₁ and (Table 1B), which makes and insensitive to the variation of CS.

At 298 K, the relative magnitude of CS compared with E + k is small, which leads to power dependencies below 1 as CS approaches 0.06 s⁻¹. Compared to lower temperatures, the difference between the power dependency of J_{eqn (16)} and J_sim is larger. The underestimated sensitivity of J_{eqn (16)} to CS originates from the neglection of the evaporation term in its derivation (i.e., the simplification of eqn (6) to eqn (10)). When cluster evaporation is significant, the backward fluxes from larger to smaller clusters kinetically limit cluster growth and make the clusters more likely to be scavenged by the pre-existing particles, but this effect is not incorporated in the derivation of eqn (12).

4 Application to atmospheric NPF data

We now apply eqn (12) to analyse atmospheric NPF data obtained from long-term measurements in urban Beijing. NPF events during this campaign were shown to be initiated by the formation of SA-DMA clusters. Details on this dataset and the long-term measurements were reported in previous studies.^8,34 Here we classify the measurement periods into spring, autumn, and winter, for which the instruments were calibrated separately. The DMA concentration is assumed to follow [DMA] = 1.8 ppt × CS/(0.017 s⁻¹) when it was not available from direct measurements, where 1.8 ppt and 0.017 s⁻¹ are the measured [DMA] and CS during NPF periods, respectively. This relationship between [DMA] and CS was based on the good correlation between the two in urban Beijing.⁸ The evaporation rate of SA₁DMA₁ (E₂) as a function of temperature was calculated from a temperature-dependent standard Gibbs free energy change. Here we use the free energy change reported in Cai et al.,⁸ which was obtained in an effort to reconcile models and observations (−14.0 kJ mol⁻¹ at 298 K). This value is within the range of data from different sources,^28–30 though it is slightly different from the value used above.

The NPF rates for this campaign were reported at a mobility size of 1.4 nm, which approximately corresponds to clusters containing 4 SA molecules. This allows us to use equations based on pathways similar to Fig. 3 to calculate the NPF rates. Previously, Cai et al.⁸ developed a detailed kinetic model to interpret the observations. Unlike the assumption we made in Fig. 3 that the addition of both SA₁ and SA₁DMA₁ leads to particle growth, their kinetic model assumes that SA₂DMA₂ and SA₃DMA₃ can only grow via the addition of SA₁DMA₁. A survey of the available cluster thermodynamics data from different sources^28–30 shows that the stability of both SA₃DMA₂ and SA₄DMA₃ against SA evaporation is uncertain (see Table S1† for a comparison of the evaporation rates of these clusters). In light of this uncertainty, we consider NPF rates for three cases: (1) both SA₃DMA₂ and SA₄DMA₃ are stable, i.e., the same NPF mechanism as that in Fig. 3, (2) the cluster SA₃DMA₂ is stable but SA₄DMA₃ is so unstable that an SA molecule instantly evaporates from the cluster, and (3) both SA₄DMA₃ and SA₃DMA₂ instantly evaporate. The expression for the NPF rate for case 1 is the same as eqn (15), while the NPF rates for cases 2 and 3 are expressed as

andrespectively. Both eqn (18) and (19) are derived from eqn (12) (the derivation is presented in the ESI†).

Fig. 7 shows the comparison between the measured NPF rates and the theoretical NPF rates calculated with eqn (15), (18) and (19). The colored solid lines represent the mean values of NPF rates calculated within different ranges of CS. Considering measurement uncertainties, the measured and theoretical rates are in good agreement. Despite different assumptions regarding cluster stabilities, eqn (15), (18) and (19) well capture the dependency of J on CS, as indicated by the good consistency among data with different CSs (i.e., colored solid lines corresponding to different CS ranges are intermingled without showing clear trends). Eqn (15), (18) and (19) contain the term to the power of 0–2, respectively, suggesting different CS dependencies. However, CS₁ is comparable to k₀ (0.025–0.1 s⁻¹) and smaller than E₁, which makes the value of a weak function of CS₁. As a result, the three equations do not show evident differences of CS-dependency, i.e., the CS-dependency of NPF rates in Beijing is mainly governed by the term . To demonstrate that this is indeed the case, Fig. S5 in the ESI† compares the CS power dependencies of and , indicating that the power dependency of the former is mostly in the range [-0.03, −0.16], while the latter is in the range [−1.1, −1.9] (numbers in brackets correspond to the 10th and the 90th percentile).

Fig. 7 Comparison of the measured and theoretical NPF rates. Data points are colored by their respective CS values during NPF measurements. The colored solid lines show the mean values of NPF rates calculated within different ranges of CS. The solid black lines correspond to J_sim:J_meas = 1 : 1 and the dashed black lines correspond to J_theo : J_meas = 1 : 10 and J_theo : J_meas = 10 : 1.

The deviations of the theoretical NPF rates from the measured values provide hints for cluster stability since the differences among eqn (15), (18) and (19) lie in the stability of SA₃DMA₂ and SA₄DMA₃. We use R² values calculated with the logarithmic of the NPF rates as a measure for the deviations. In agreement with the visual inspection of Fig. 7, eqn (18) better predicts NPF with an overall R² value of 0.42, slightly better than R² = 0.34 given by eqn (15) and (19). This could indicate that SA₃DMA₂ and SA₄DMA₃ are not both stable, although the somewhat arbitrary free energy of SA₁DMA₁ (see above) and measurement uncertainties prevent us from drawing a definitive conclusion. As a reference, the R² value based on the detailed model of Cai et al.⁸ is 0.42.

Eqn (15), (18) and (19) were derived from eqn (12) without solving the systems of cluster population balance equations. This was enabled by the simplifying assumption that the evaporation term on the R.H.S. of eqn (6) is negligible. Consequently, each cluster formation step in the nucleation pathway simply corresponds to a multiplicative term in the NPF rate expression in the form of (with necessary modifications). In future studies, we aim to apply the developed framework to other NPF mechanisms.

5 Conclusions

We developed analytical equations for NPF rates in the presence of pre-existing particles for nonbranched nucleation pathways. When clusters do not evaporate, by comparison with simulations it was shown that the proposed equations can predict the NPF rates with biases less than a factor of 2 for J₄ and J₆ even though the equations do not include particle coagulation. The proposed equations also correctly describe the dependency of the NPF rates on the nucleating vapor concentration. A high CS leads to strong deviations of NPF rates from the collision-controlled limit, the extent of which depends on the size at which NPF rates are measured.

When clusters do evaporate, a simplified expression for the NPF rates (i.e., eqn (12)) was derived by neglecting the backward cluster fluxes in the cluster balance equation. Our analysis shows that it is the multiplicative term that determines the NPF rates. CS and E work synergistically to suppress nucleation, and this synergy is embodied in the cumulative product . We compared the variants of eqn (12) to benchmark cluster dynamics simulations of SA-DMA nucleation and SA-AM nucleation. In the former system cluster evaporation is slow, while in the latter system cluster evaporation is more significant. We found that for both test chemistries the developed equations give fair predictions of the NPF rates with errors less than a factor of 3 under the test conditions. Additionally, the developed equations approximately capture the power dependency of the NPF rates on CS. However, when cluster evaporation rates are high and the underlying assumption of eqn (12) is violated (e.g., the SA-AM nucleation at 298 K), it was found that the theoretical NPF rates are less sensitive to CS than predicted by cluster dynamics simulations.

We applied the theoretical equations to calculate atmospheric NPF rates. The theoretical NPF rates were found in agreement with field NPF measurements during a long-term campaign in Beijing. Other potential applications of the theoretical equations include NPF by the iodine species¹⁷ and cluster growth by HOM condensation. To model the latter system, the many species of HOMs may need to be grouped by volatility, with the cluster growth rates and evaporation rates calculated with group-averaged properties.³⁵ A scenario not considered in this work is ion-mediated nucleation, in which the coagulation rates of oppositely charged clusters are greatly enhanced by Coulomb interactions; further assessment of the applicability of the proposed method to such scenarios is needed.

The focus of this work is the proposal and verification of a simplified method to incorporate CS into analytical expressions for atmospherically relevant NPF systems. For a given NPF mechanism, more sophisticated models can be built, and cluster dynamics simulations can be used to calculate the NPF rates.^8,9,36 Both these methods should in theory be more accurate than the approximate equations (i.e., eqn (12) and variants) used in this work. However, eqn (12) offers physically intuitive descriptions of the interactions between cluster growth, evaporation, and CS without sacrificing much accuracy: the proportion of any cluster that can eventually become nucleated particles depends on the ratio of the forward flux (cluster growth) to the sum of all fluxes (cluster growth + loss to pre-existing particles + cluster evaporation).

Author contributions

CL and HX initialized the study. CL developed the equations and did the cluster dynamics simulations. LL and XZ supported the study with quantum chemical simulation data. JZ, VK, MK, JK and RC supported the study with field measurements and data analysis. CL and RC took the lead in writing the manuscript and the other authors contributed to the writing and revision of the manuscript.

Conflicts of interest

There are no conflicts of interest to declare.

Acknowledgements

This work was supported by the Natural Science Foundation of Shanghai (grant no. 21ZR1430100), Shanghai Sailing Program (grant no. 20YF1418600), Academy of Finland (project no. 332547), National Natural Science Foundation of China (grant no. 22188102) and Samsung PM2.5 SRP.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d2ea00066k

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