Gas-phase aldol condensation of formaldehyde to produce hydroxyacetaldehyde and its implication to new particle formation: a theoretical study

Abstract

Aldehydes have been proposed as important precursor species in new particle formation (NPF). Although formaldehyde (CH₂O) has minimal direct involvement in sulfuric acid (H₂SO₄) and water nucleation, it remains unclear whether its atmospheric aldol condensation product, hydroxyacetaldehyde (C₂H₄O₂), one of the simplest bifunctional oxygenated volatile organic compounds (OVOCs), plays a role in NPF. This study investigates both the aldol condensation of CH₂O and its role in NPF involving H₂SO₄ and C₂H₄O₂ through quantum chemical calculations and atmospheric cluster dynamics modeling. Kinetic calculations indicate that the reaction rate of CH₂O aldol condensation catalyzed by H₂SO₄ is 8 to 16 orders of magnitude higher than that of the uncatalyzed pathway at 200–298 K. Based on molecular structures and formation Gibbs free energies, interactions between sulfuric acid/its polymers and C₂H₄O₂ are thermodynamically favorable. Furthermore, C₂H₄O₂, with its hydroxyl group, stabilizes H₂SO₄ clusters more effectively than CH₂O, thereby enhancing nucleation. Additional cluster kinetic modeling suggests that particle formation rates in this system exceed those in the sulfuric acid–water binary system under conditions of low ambient H₂SO₄ concentrations and low relative humidity. However, cluster growth remains limited due to weak formation of larger clusters, indicating that other stabilizing vapors are needed for sustained cluster growth and stable particle formation.

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1. Introduction

The formation of atmospheric particles is of great interest because of the atmospheric particles' impact on global climate and health.^1–3 New particle formation (NPF) accounts for over 50% of atmospheric cloud condensation nuclei (CCN), significantly impacting cloud properties and Earth's energy balance.⁴ Despite this importance, the chemical identity and relative significance of participating vapors remain insufficiently understood. Sulfuric acid (H₂SO₄) is widely recognized as a critical nucleation precursor in the atmosphere due to its extremely low volatility and high acidity.^5,6 However, both sulfuric acid–water (H₂SO₄–H₂O) binary nucleation and ternary nucleation involving ammonia or amines fail to fully explain observed NPF events under the complex and varied conditions of the atmosphere,^7–12 indicating other chemical species may also play important roles in NPF.^6,13,14

Formaldehyde (CH₂O) is a compound of significant interest due to its critical role as an intermediate in atmospheric photochemical reactions, where it notably enhances both atmospheric reactivity and oxidative capacity. Furthermore, given its wide range of applications, exposure to CH₂O is linked to both acute and chronic health effects.¹⁵ Its atmospheric concentrations vary widely, ranging from several thousand pptv to tens of ppbv, depending on geographic and environmental factors.^16–18 Budget analyses of CH₂O show significant discrepancies between observed concentrations and those predicted by models.^19,20 This has prompted increased interest in alternative CH₂O removal mechanisms, including uptake by soil surfaces,²¹ aerosols/clouds,^22–24 as well as its direct participation in nucleation processes.^25–27 These pathways may represent additional sinks for CH₂O, offering potential explanations for the overestimation of its concentrations in atmospheric models. Our previous flow tube experiments indicate that the enhancement of CH₂O in H₂SO₄–H₂O homogeneous nucleation is negligible.²⁶ However, we also discovered that the hydrolysis product of CH₂O, methanediol (CH₂(OH)₂), forms hydrogen bonds with sulfuric acid and its polymer through its hydroxyl groups, thereby contributing to cluster stabilization.²⁸ The possible product of atmospheric formaldehyde via aldol condensation is hydroxyacetaldehyde (glycolaldehyde, C₂H₄O₂).²⁹ Despite this, the kinetics of CH₂O aldol condensation remain insufficiently understood, and its potential involvement in NPF is unknown. Previous studies have demonstrated that the products of aldehyde aldol condensation can contribute to particle growth in the atmosphere.^25,30,31 For instance, Shi et al.²⁵ found that atmospheric aldol condensation products of aldehydes are more likely to form clusters with sulfuric acid than the aldehydes themselves, as demonstrated by quantum chemical calculations. Therefore, it is systematic and meaningful to further investigate the impact of C₂H₄O₂ on NPF.

As one of the simplest bifunctional oxygenated volatile organic compounds (OVOCs), C₂H₄O₂ contains both aldehyde and alcohol functional groups. It is primarily produced from isoprene³² (468 Tg C per year, with 22% gas phase conversion to C₂H₄O₂ via hydroxyl radicals³³) and ethene,³⁴ and has been detected in biomass burning plumes.³⁵ Its ambient concentration is approximately one-seventh that of formaldehyde,³⁶ indicating that the concentration of C₂H₄O₂ ranges from hundreds of pptv to several ppbv in various regions of the world. In the Southeastern U.S.A., measurements over a forested areas below 2 km altitude have recorded levels of up to 3 ppbv, with a mean concentration of 1 ppbv.³⁷ In addition to structural hydroxyl content and high concentration, indicating the possible involvement in NPF, C₂H₄O₂ also exhibits a relatively low vapor pressure. Estimated at 8.32 × 10⁵ atm at 298 K, this vapor pressure is notably lower than that of CH₂O (i.e. 5.13 atm).²⁹ Due to this reduced volatility, C₂H₄O₂ molecules are more prone to condensation, thereby contributing to atmospheric particle growth. Bulk aqueous hydroxyl radicals with C₂H₄O₂ experiments performed by Perri et al.³⁸ show that C₂H₄O₂ (as well as glyoxal and methylglyoxal) is an important source of secondary organic aerosol. Recent research indicates that the decomposition of C₂H₄O₂ could represent a significant initial step in new particle formation, based on thermodynamic calculations.³⁹ Nevertheless, further kinetics investigation including collision and evaporation rate is required to delineate the mechanism of new particle formation.

In this study, we investigate the catalytic effect of sulfuric acid (SA) in the aldol condensation of CH₂O to produce C₂H₄O₂ and conduct a comparative analysis of molecular cluster formation between SA and CH₂O, as well as SA and C₂H₄O₂, using a combination of quantum chemical calculations and kinetic modeling via the Atmospheric Cluster Dynamics Code (ACDC).^40,41 After performing systematic conformational searches, we obtained minimum Gibbs free energy structures of clusters with compositions (SA)_m(B)_n (0 ≤ m, n ≤ 3; “B” represents CH₂O or C₂H₄O₂). The corresponding thermodynamic data were then applied in ACDC to obtain particle formation rates. Furthermore, the effect of the vapor concentrations on cluster formation was assessed.

2. Methods

2.1. Quantum chemical calculations

The basin-hopping (BH) algorithm^42–44 coupled with the PM7 semiempirical potential⁴⁵ implemented in the MOPAC 2016 program (https://openmopac.net) was employed to search for the initial (SA)_m(B)_n (0 ≤ m, n ≤ 3; “B” represents CH₂O or C₂H₄O₂) geometries, which is similar to our previous studies.^28,46–48 Then, the top 20 lowest-lying conformers of each clusters were optimized at the PW91PW91/6-311++G(3df,3pd) level to determine the final configurations with the Gaussian 09 software package.⁴⁹ Harmonic vibrational frequencies were calculated to confirm that these obtained conformers were the true minima. The method provides good geometries,^50,51 excellent vibrational frequencies⁵² and quite accurate cluster Gibbs free energies compared with the currently available experiments.^53–55 Benchmark details of the methods employed in atmospheric cluster calculations can be found in our previous study.²⁶

2.2. Kinetics calculations

For kinetic calculations, geometry optimization of all reactants, prereaction complexes, transition states, postreaction complexes, and products were performed using M06-2X⁵⁶ functional at the 6-311++G(d,p)⁵⁷ basis set with the Gaussian 09 software package. Furthermore, intrinsic reaction coordinate (IRC) calculations⁵⁸ were performed at the same level to determine whether the located transition states connect with the desired reactants and products. In addition, to refine the relative energies of the various stationary points, single-point energy calculations were carried out at the DLPNO-CCSD(T)/aug-cc-pVTZ level of theory with the ORCA 4.0 suite of programs.⁵⁹ To evaluate the effects of SA on the rate constants of the gas-phase aldol condensation of CH₂O, conventional transition-state theory (TST) with Eckart tunneling correction^60,61 was used to calculated reaction rate constants with the KiSThelP program.⁶²

We used the ACDC to study formation rates and evaporation properties of (SA)_m(B)_n (0 ≤ m, n ≤ 3; “B” represents CH₂O or C₂H₄O₂) clusters. The code generates and solves the cluster birth-death equations, the time derivatives of the concentrations of all constituents included in the simulation, which essentially is a series of logical checks over all possible cluster combinations to see which evaporations and collisions can create or destroy a given cluster.^41,63,64 The code generates and solves the cluster birth-death equations, the time derivatives of the concentrations of all constituents included in the simulation as eqn (1):

where c_i is the concentration of cluster i, β_i,j is the collision coeffcient of clusters i with j, and γ_{i + j→j} is the evaporation coeffcient of cluster i + j evaporating into clusters i and j. Q_i is the possible additional source of cluster i and S_i is the sink term of cluster i. The collision rate constants were calculated from the kinetic gas theory, and the evaporation rate constants were calculated from the Gibbs free energies of formation of the clusters according to the concept of detailed balance.where c^e_i is the equilibrium concentration of cluster i, ΔG_i is the Gibbs free energy of the formation of cluster i, and c_ref is the monomer concentration of the reference vapor corresponding to the pressure of 1 atm at which the Gibbs free energies were determined.

In addition, the cluster formation rate in our study is defined as the flux of clusters outside the “3 × 3 box” system, where 3 is the maximum number of H₂SO_4, CH₂O or C₂H₄O₂ in the clusters, assuming the clusters on the boundaries are large enough to have negligible evaporation coefficients, since these clusters are not allowed to re-enter, it is as if they have become stable particles. A constant coagulation sink coefficient (sink term) and the source rate was set to zero for simplicity.

3. Results and discussion

3.1. Aldol condensation of formaldehyde without/with catalyzed sulfuric acid

Given the estimated standard formation Gibbs free energy (ΔG⁰_f) of CH₂O and β-hydroxycarbonyl, C₂H₄O₂, summarized by Barsanti and Pankow (−102.5 and −270.4 kJ mol⁻¹),²⁹ the standard Gibbs free energies changes (ΔG⁰) for formaldehyde aldol condensation is −15.63 kcal mol⁻¹ according to the fundamental eqn (3), indicating the reaction is favorable.

The formation of C₂H₄O₂ via aldol condensation can be expressed as

CH₂O + CH₂O ⇌ C₂H₄O₂ (4)

There exist numerous reports on atmospherically proton transfer reactions and sulfuric acid can act as relatively strong hydrogen-atom donors/acceptors,^65–67 thereby possibly catalyzing the aldol condensation of CH₂O. In the presence of the catalyst H₂SO₄, they are trimolecular reaction systems, the reaction proceeds via collision with each other to form dimers, and then the dimers encounter the third reactant to form the (CH₂O)₂(H₂SO₄) complex, which is followed by unimolecular transformation to form (C₂H₄O₂)(H₂SO₄) complex in the exit channel. There are two possible entrance channels in the reaction of CH₂O + CH₂O + H₂SO₄ → C₂H₄O₂ + H₂SO₄, and they can be expressed as

H₂SO₄ + CH₂O ⇌ (H₂SO₄)(CH₂O) (5)

CH₂O + (H₂SO₄)(CH₂O) → C₂H₄O₂ + H₂SO₄ (6)

and

CH₂O + CH₂O ⇌ (CH₂O)₂ (7)

(CH₂O)₂ + H₂SO₄ → C₂H₄O₂ + H₂SO₄ (8)

From the perspective of spatial structure, (H₂SO₄)(CH₂O) reacts more readily, and the binding energy of the (H₂SO₄)(CH₂O) (−10.6 kcal mol⁻¹) is lower than that of (CH₂O)₂ (−2.6 kcal mol⁻¹) as shown in Fig. 1. Therefore, the (H₂SO₄)(CH₂O) + CH₂O entrance channel is considered here.

Fig. 1 Potential electronic energy surfaces with zero-point vibrational energies corrected at the DLPNO-CCSD(T)/aug-cc-pVTZ//M06-2X/6-311++G(d,p) level of theory (in kcal mol⁻¹) for the reaction of CH₂O + CH₂O with (a) no catalyst and (b) H₂SO₄ as a catalyst.

For the direct aldol condensation of CH₂O without a catalyst reaction (Fig. 1a), the fairly high reaction barrier resulting in the reaction is not a plausible path. The energy barrier is calculated to be 78.4 kcal mol⁻¹ with respect to the prereactive complex and a large ring tension of the rather closed four-membered ring is in the transition state (TS1) geometry, making the path kinetically unfavorable. For the reaction catalyzed by H₂SO₄ (Fig. 1b), the energy barrier is calculated to be 39.2 kcal mol⁻¹ with respect to the reactants CH₂O and (H₂SO₄)(CH₂O), and it is clear that the reaction energy barrier of CH₂O aldol condensation catalyzed by H₂SO₄ is reduced by 39.2 kcal, which indicates that the H₂SO₄ exerts a strong catalytic effect. However, the relatively high reaction barrier causes the reaction to be difficult to occur as well.

The atmospheric implications of the reactions studied would be determined by how fast different reaction channels are and the competition between them, so the reaction rate coefficients are calculated here. Applying the steady-state approximation to the prereactive complex and assuming that the complex is in equilibrium with the reactant, similar to the formation of H₂SO₄ (ref. 68) and organic nitrate,⁶⁹ the H₂SO₄-catalyzed formation reaction rate of C₂H₄O₂ (v_H2SO4) can be described as eqn (9):

k₄ represents the bimolecular rate constants of the (H₂SO₄)(CH₂O) + CH₂O reaction, which has been calculated using conventional transition-state theory with Eckart tunneling. The overall rate constant of the H₂SO₄-catalyzed formaldehyde aldol condensation (k_{eff, H2SO4}) is represented by eqn (8):

k_{eff, H2SO4} = K_{(H2SO4)(CH2O)} × k₄ × [H₂SO₄] (10)

The rate constants for each channel and equilibrium constants over the temperature range of 200–298 K are presented in Table 1. Without considering the catalyst, the rate constant k_un is 9.84 × 10⁻⁸¹ to 6.52 × 10⁻⁶⁴ cm³ per molecule per s at 200–298 K, which is too small for the reaction to occur. In the H₂SO₄ catalytic channel, the reaction rate constant is 2.23 × 10⁻⁶⁴ to 2.01 × 10⁻⁵⁵ cm³ per molecule per s at a typical atmospheric [H₂SO₄] of 10⁷ molecules per cm³, where the H₂SO₄ concentrations in the atmosphere span a wide range from 10⁴ to 10⁹ molecules per cm³.^70,71 To obtain a more complete knowledge of the sulfuric acid effect in the CH₂O hydrolysis reaction, it is necessary to compare the rate of the naked and sulfuric acid-assisted reactions rather than comparing the reaction energy barriers or the rate constants of the individual reactions. The rate ratio, v_eff/v_un, listed in Table 1 shows that the reaction with the H₂SO₄ catalytic channel is 8 to 16 orders of magnitude faster than the reaction without H₂SO₄. As a result, H₂SO₄ could efficiently make the CH₂O aldol condensation process more feasible than the uncatalyzed channel both energetically and kinetically.

Table 1 Equilibrium constants (K_{(H2SO4)(CH2O)}, cm³ per molecules), the reaction rate coefficients (k, cm³ per molecule per s), and the reaction rates (v, molecules per cm³ per s) for the formation of C₂H₄O₂ without/with catalyzed sulfuric acid between 200 and 298 K calculated at the DLPNO-CCSD(T)/aug-cc-pVTZ//M06-2X/6-311++G(d,p) level of theory

M	200 K	220 K	240 K	260 K	280 K	298 K
a The rate constant of the reaction of CH2O + (H2SO4)(CH2O).b The rate constant of the reaction of CH2O + CH2O.c [H2SO4] is 10^7 molecules per cm^3.d The relative rate ratio between CH2O + CH2O + H2SO4 and CH2O + CH2O when [H2SO4] is 10^7 molecules per cm^3.
K(H2SO4)(CH2O)	2.31 × 10^−15	2.15 × 10^−16	3.01 × 10^−17	5.77 × 10^−18	1.42 × 10^−18	4.74 × 10^−19
k6^a	9.65 × 10^−57	1.70 × 10^−53	1.34 × 10^−50	4.57 × 10^−48	7.44 × 10^−46	4.24 × 10^−44
kun^b	9.88 × 10^−81	3.94 × 10^−76	2.56 × 10^−72	5.20 × 10^−69	3.74 × 10^−66	6.55 × 10^−64
keff^c	2.23 × 10^−64	3.65 × 10^−62	4.03 × 10^−60	2.64 × 10^−58	1.05 × 10^−56	2.01 × 10^−55
veff/vun^d	2.26 × 10^16	9.28 × 10^13	1.57 × 10^12	5.08 × 10^10	2.82 × 10^9	3.07 × 10^8

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3.2. Structures and thermodynamic analysis

To elucidate the effect of C₂H₄O₂ on sulfuric acid nucleation, structures and thermodynamic values of (H₂SO₄)_m(C₂H₄O₂)_n (0 ≤ m, n ≤ 3) clusters were discussed in this section. The cluster formation steps with optimized structures of global Gibbs free energy minima are shown in Fig. 2, and corresponding enthalpies and entropies for (H₂SO₄)_m(C₂H₄O₂)_n (0 ≤ m, n ≤ 3) clusters formation are shown in Table S1 in the (ESI).† Here, (H₂SO₄)_m(C₂H₄O₂)_n are abbreviated as mAnB and the values in parentheses are the binding Gibbs free energies calculated using the following equation:

ΔG((SA)_m(C₂H₄O₂)_n) = G((SA)_m(C₂H₄O₂)_n) − m × G_SA − n × G_C2H4O2 (11)

Fig. 2 Diagram for the studied cluster formation steps, with structures of global Gibbs free energy minima in the (H₂SO₄)_m(C₂H₄O₂)_n (m = 1–3, n = 1–3) system obtained at the PW91PW91/6-311++G(3df,3pd) level of theory. All presented values are the calculated Gibbs free energy changes and the values in parentheses are the binding Gibbs free energies in kcal mol⁻¹ at 298.15 K and 1 atm.

The cluster formation between a SA and a C₂H₄O₂ molecule involves the formation of one hydrogen bond as shown in Fig. 2. The reaction Gibbs free energy for forming (H₂SO₄)(C₂H₄O₂) cluster is found to be −1.78 kcal mol⁻¹. This process is more favorable than the formation of the (H₂SO₄)(CH₂O) complex, with a Gibbs free energy change of −1.15 kcal mol⁻¹, as shown in Table 2. However, it is slightly less favorable than the formation of the (H₂SO₄)(H₂O), with a ΔG of −1.94 kcal mol⁻¹, and significantly less favorable than the formation of H₂SO₄ dimer and (H₂SO₄)(NH₃). This result infers that β-hydroxycarbonyl is stronger for stabilizing SA to promote atmospheric particle nucleation than the simple aldehyde, however, the interaction of β-hydroxycarbonyl with SA is still weaker than that of ammonia.

Table 2 Enthalpies, entropies, and Gibbs free energies changes associated with the affinities of hydroxyacetaldehyde/formaldehyde to monomers, dimers and trimers of sulfuric acid calculated at the PW91PW91/6-311++G(3df,3pd) level of theory at 298.15 K and 1 atm

Reactions	ΔH (kcal mol^−1)	ΔS (cal mol^−1 K^−1)	ΔG (kcal mol^−1)
H2SO4 + C2H4O2 ⇔ (H2SO4)(C2H4O2)	−11.16	−31.47	−1.78
H2SO4 + CH2O ⇔ (H2SO4)(CH2O)	−10.34	−30.82	−1.15
H2SO4 + H2O ⇔ (H2SO4)(H2O)	−11.84	−33.19	−1.94
H2SO4 + NH3 ⇔ (H2SO4)(NH3)	−20.82	−31.96	−11.29
H2SO4 + H2SO4 ⇔ (H2SO4)2	−16.54	−37.49	−5.36
(H2SO4)2 + C2H4O2 ⇔ (H2SO4)2(C2H4O2)	−13.91	−37.51	−2.72
(H2SO4)2 + CH2O ⇔ (H2SO4)2(CH2O)	−10.65	−29.80	−1.77
(H2SO4)2 + H2O ⇔ (H2SO4)2(H2O)	−14.28	−36.56	−3.38
(H2SO4)2 + NH3 ⇔ (H2SO4)2(NH3)	−29.20	−40.85	−17.02
(H2SO4)2 + H2SO4 ⇔ (H2SO4)3	−15.19	−42.02	−2.66
(H2SO4)3 + C2H4O2 ⇔ (H2SO4)3(C2H4O2)	−15.46	−36.17	−4.67
(H2SO4)3 + CH2O ⇔ (H2SO4)3(CH2O)	−13.86	−34.39	−3.60
(H2SO4)3 + H2O ⇔ (H2SO4)3(H2O)	−17.48	−37.57	−6.28
(H2SO4)3 + NH3 ⇔ (H2SO4)3(NH3)	−30.78	−32.88	−20.97
(H2SO4)3 + H2SO4 ⇔ (H2SO4)4	−14.73	−42.27	−2.13

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The two subsequent additions of sulfuric acid molecules to the (H₂SO₄)(C₂H₄O₂) complex via the formation of SA–SA hydrogen bonded interactions. These processes are found to be more favorable (i.e. −6.31 and −4.60 kcal mol⁻¹) compared to the first H₂SO₄ addition, which is due that the addition of a H₂SO₄ molecule leads to a more reduction in the enthalpy though the clustering process is accompanied by an entropy decrease⁷ as the formation of hydrogen bonds leads to a more constrained structure. From the molecular structures of these clusters, it is apparent that the interactions strength are as follows: sulfuric acid-sulfuric acid > β-hydroxycarbonyl-sulfuric acid > β-hydroxycarbonyl-β-hydroxycarbonyl.

The (H₂SO₄)₂(C₂H₄O₂)₂ cluster formation via adding a H₂SO₄ molecule to (SA)(C₂H₄O₂)₂ is more favorable than adding a C₂H₄O₂ molecule to (H₂SO₄)₂(C₂H₄O₂). However, because the formation of (H₂SO₄)₂(C₂H₄O₂) is more favorable than the formation of (H₂SO₄)(C₂H₄O₂)₂, where the corresponding Gibbs free energy changes from (H₂SO₄)₂(C₂H₄O₂) are −6.31 and −2.61 kcal mol⁻¹, respectively, the (H₂SO₄)₂(C₂H₄O₂)₂ cluster would be formed along (H₂SO₄)(C₂H₄O₂) → (H₂SO₄)₂(C₂H₄O₂) → (H₂SO₄)₂(C₂H₄O₂)₂ path. The (H₂SO₄)₃(C₂H₄O₂)₃ cluster formation shows this feature as well, with the path along (H₂SO₄)(C₂H₄O₂) → (H₂SO₄)₂(C₂H₄O₂) → (H₂SO₄)₃(C₂H₄O₂) → (H₂SO₄)₃(C₂H₄O₂)₂ → (H₂SO₄)₃(C₂H₄O₂)₃. Looking to the molecular structures of these clusters, the pattern is also due to the different interaction strength levels between molecules. The above pathways are not unique due that the channels are competing with each other and there is an actual channel occupancy (growth flux) considering the cluster evaporation rate. In the following simulations of steady-state formation rates, growth fluxes were considered.

Comparing the affinity of C₂H₄O₂ and CH₂O to dimers and trimers of sulfuric acid as shown in Table 2, similar to the reaction between C₂H₄O₂ and H₂SO₄ molecule, the C₂H₄O₂ affinity to dimers/trimers of sulfuric acid, with a value of −4.73/−6.67 kcal mol⁻¹, is higher than that the CH₂O affinity, however, it is much less than the corresponding ammonia affinity. In addition, the C₂H₄O₂ affinities to sulfuric acid dimer and sulfuric acid trimer are higher than that sulfuric acid itself with a value of −2.66 and −2.13 kcal mol⁻¹, respectively. Here, it once again proves that β-hydroxycarbonyl is stronger for stabilizing sulfuric acid and its polymer to promote nucleation than simple aldehydes. In conclusion, aldol condensation of CH₂O can apparently enhance the binding strength with the atmospheric nucleation precursor of sulfuric acid and its polymer by introducing a functional hydroxyl group. CH₂O and its atmospheric derivatives are unlikely to be key species directly involved in nucleation, such as ammonia and amines. All optimized Cartesian coordinates of (H₂SO₄)_m(C₂H₄O₂)_n (0 ≤ m, n ≤ 3), (H₂SO₄)_m(H₂O) (0 ≤ m ≤ 3) and (H₂SO₄)_m(NH₃) (0 ≤ m ≤ 3) discussed here are shown in Table S2 in the ESI.† The optimized structures and Cartesian coordinates for (H₂SO₄)_m(CH₂O)_n (0 ≤ m, n ≤ 3) clusters be found in our previous study.²⁸

3.3. Kinetics analysis

3.3.1. Cluster Gibbs free energy surfaces. In order to check whether the nucleation barrier is high or not or maybe nonexistent, the Gibbs free energy change, ΔG, should be converted into the actual Gibbs free energy change,⁷²where k_B is the Boltzmann constant, T is the temperature, P_ref is the reference pressure (1 atm in this case), N₁ and N₂ is the molecule number in the cluster for composition 1 and 2, respectively, and P₁ and P₂ is the partial pressure of component 1 and component 2 in vapor phase, respectively. Here, 1 and 2 means SA and B (CH₂O or C₂H₄O₂). Fig. 3 shows the actual formation Gibbs free energy surface (in kcal mol⁻¹) on the SA–B grid at 278.15 K, SA concentration of 10⁷ molecules per cm³, and 1000 pptv of B.

Fig. 3 Gibbs free energy surface of the H₂SO₄–CH₂O system in figure (a) and H₂SO₄–C₂H₄O₂ system in figure (b) calculated with PW91PW91/6-311++G(3df,3pd) at 278.15 K, [H₂SO₄] = 10⁷ molecules per cm³, and [B] = 1000 pptv.

While the absolute value of the formation Gibbs free energies varies between the different systems, a similar trend is seen for the two cases, with a large Gibbs free energy barrier in all directions for forming larger clusters. For any given cluster there is no growth direction that leads to a lower formation Gibbs free energy via addition of either B or SA. Following the path with lowest Gibbs free energy from (SA)(B) cluster, the order of growth for the H₂SO₄–C₂H₄O₂ systems is continuous addition of a C₂H₄O₂ molecule, which is different from the cluster formation steps only considering Gibbs free energy change as shown in Fig. 3, while, the order of growth for the H₂SO₄–CH₂O system is firstly addition of a H₂SO₄ molecule followed by addition of a CH₂O molecule. In general, it is seen that the Gibbs free energy steeply increases towards the system boundaries for these two cases, which implies that the growth within the system is unfavorable.

3.3.2. Evaporation rates. The competition between the forward reaction by adding a molecule and the reverse reaction by evaporation at each intermediate step determines whether a cluster grows to form a nanoparticle, and the collision and evaporation rates can be used to infer the stability of clusters. The total evaporation rates for the (H₂SO₄)_m(B)_n (0 ≤ m, n ≤ 3; “B” represents CH₂O or C₂H₄O₂) clusters on the H₂SO₄–B grid at 278.15 K are shown in Fig. 4. While the evaporation rates for clusters vary between the different systems, the evaporation rates for most (H₂SO₄)_m(C₂H₄O₂)_n (0 ≤ m, n ≤ 3) clusters are lower than those for the (H₂SO₄)_m(CH₂O)_n (0 ≤ m, n ≤ 3) clusters. This indicates that (H₂SO₄)_m(C₂H₄O₂)_n (0 ≤ m, n ≤ 3) clusters are more stable than those CH₂O-involved clusters. In H₂SO₄–C₂H₄O₂ system, clusters with a higher number of H₂SO₄ than C₂H₄O₂ molecules have lower evaporation rates, for example, evaporation rates of (H₂SO₄)₂(C₂H₄O₂)₁ and (H₂SO₄)₃(C₂H₄O₂)₂ are smaller than those of (H₂SO₄)₁(C₂H₄O₂)₂ and (H₂SO₄)₂(C₂H₄O₂)₃, respectively. Therefore, clusters with a higher number of H₂SO₄ than C₂H₄O₂ molecules are more stable. In general, the total evaporation rates for the H₂SO₄–C₂H₄O₂ and H₂SO₄–CH₂O system are high. Therefore, it is unlikely that B and SA by themselves drive new particle formation at 278 K. In addition, higher concentrations of precursors can enhance the stability of clusters, as higher precursor concentrations increase the probability of cluster collisions and shift the balance between collision and evaporation forward. The effect of the precursors concentrations on the particle formation rate will be discussed next.

Fig. 4 The total evaporation rates for the (H₂SO₄)_m(CH₂O)_n (0 ≤ m, n ≤ 3) clusters in figure (a) and (H₂SO₄)_m(C₂H₄O₂)_n (0 ≤ m, n ≤ 3) clusters in figure (b) at the PW91PW91/6-311++G(3df,3pd) level of theory at 278.15 K.

3.3.3. Steady-state formation rates. New particles could form when the collision rate of monomers to the clusters exceed the cluster evaporation rates beyond some cluster size. Fig. 5 shows the steady-state formation rate of particles (J) growing out of the simulation systems as a function of monomer concentration at 278.15 K for H₂SO₄–CH₂O and H₂SO₄–C₂H₄O₂ systems. The simulations were performed at ambient SA concentrations starting from 10⁶ to 10⁸ molecules per cm³ and at ambient CH₂O and C₂H₄O₂ concentrations for comparing.

Fig. 5 Simulated particle formation rate J (cm⁻³ s⁻¹) out of the simulation system as a function of H₂SO₄ monomer concentration at 278.15 K with different B mixing ratios for H₂SO₄–CH₂O system in figure (a) and H₂SO₄–C₂H₄O₂ system in figure (b). The dashed black lines show the prediction calculated using the parameterized binary homogeneous nucleation of H₂SO₄–H₂O at 278.15 K and RH = 38%.

Generally, J increase with increasing the concentrations of B and H₂SO₄ at the simulated condition. The H₂SO₄ concentration dependence of the cluster formation rate does not change with CH₂O and C₂H₄O₂ concentration, with the power dependency of 3; the power dependency on C₂H₄O₂ does not change with SA concentration, with the value of about 4. J of the H₂SO₄–C₂H₄O₂ system are 7–8 orders of magnitude greater than those of the H₂SO₄–CH₂O system at the same conditions with [B] = 1000 ppt. Furthermore, J of the H₂SO₄–C₂H₄O₂ system are higher than those of H₂SO₄–H₂O binary homogeneous nucleation at 278.15 K and 38% relative humidity (RH) according to the parameterization suggested by Vehkamäki et al.⁷³ when SA concentration is less than 6 × 10⁷ molecules per cm³, indicating that the cluster formation for SA with C₂H₄O₂ is more favorable than that with water at ambient low SA concentration and low RHs.

4. Conclusions

Aldehydes were speculated as important precursor species in the NPF, and aldol condensation of formaldehyde introduces functional group of –OH and would have lower vapor pressure, hence aldol condensation product is thought to participated in atmospheric NPF. The kinetics of CH₂O aldol condensation to produce C₂H₄O₂ was examined and the potential role of C₂H₄O₂ in sulfuric acid-driven atmospheric NPF was explored. Structures and thermodynamics up to the cluster size of (H₂SO₄)₃(B)₃ are studied, and geometries and ΔG values calculated at 298.15 K and 1 atm show that the C₂H₄O₂ likely stabilize sulfuric acid and its polymer better than CH₂O.

This study is based on our previous research on NPF involving hydrolysate of formaldehyde and serves as the first kinetic investigation of clusters containing SA and aldol condensation product of formaldehyde. Particle formation rates for H₂SO₄–C₂H₄O₂ system are much higher than those for H₂SO₄–CH₂O system and higher than those of H₂SO₄–H₂O binary homogeneous nucleation at ambient low SA concentration and low RHs. However, the growth of H₂SO₄–C₂H₄O₂ clusters is essentially limited by a weak formation of the largest clusters studied. Therefore, the direct involvement of aldol condensation product of formaldehyde in sulfuric acid nucleation is negligible, and other stabilizing vapors are required in sulfuric acid-driven atmospheric NPF. In view of the rich content of carbonyls and different carbonyls show different capabilities to participate in the NPF, the role of higher aldehydes and dicarbonyls in atmospheric particle nucleation and further growth deserve study in the future.

Data availability

The data that support the findings of this study are available on request from the corresponding author, [Chunyu Wang, E-mail: cyw2022@chu.edu.cn], upon reasonable request.

Author contributions

NT and LZ performed investigation, calculations, visualization, analyzed the data and wrote original draft; CW instructed the design of the algorithm, performed investigation, and revised the manuscript; JC and YP helped to interpret the results and revised the manuscript; HX helped analyzed the data. All authors contributed to the manuscript preparation and approved the submitted version.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (grant no. 42107112), Natural Science Foundation of Anhui Province (grant no. 2308085QD130), Provincial Undergraduate Training Program for Innovation and Entrepreneurship (S202310380105), Key Scientific Research Project of Anhui Education Department (2022AH051712) and Chaohu University for the Start-Up grant (KYQD-202215).

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Footnotes

† Electronic supplementary information (ESI) available: Cluster formation energies from monomers for (H₂SO₄)_m(C₂H₄O₂)_n (0 ≤ m, n ≤ 3) and all Cartesian coordinates of optimized global Gibbs free energy minima for (SA)_m(C₂H₄O₂)_n (0 ≤ m, n ≤ 3), (H₂SO₄)_m(NH₃) (0 ≤ m ≤ 3), and (H₂SO₄)_m(H₂O) (0 ≤ m ≤ 3). See DOI: https://doi.org/10.1039/d4ra08063g

‡ These authors contributed equally to this work.

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