The gaseous hydrolysis of HNSO₂ with MSA was comprehensively studied through quantum chemistry simulations. Optimization of all the species was carried out by using the method of M06-2X with a 6-311++G(2df,2pd) basis set (Zhao and Truhlar, 2008; Elm et al., 2012; Bork et al., 2014b). Vibrational frequencies were subsequently computed at the M06-2X/6-311++G(2df,2pd) level to ensure the reality of all stationary points' frequencies and the presence of only one imaginary frequency in transition states. It is noted that the calculated bond distances and bond angles at the M06-2X/6-311++G(2df,2pd) level (Fig. S2) agree well with the available values (Fig. S2) from the experiment and three different theoretical levels of the M06-2X/6-311++G(3df,2pd), M062X/6-311++G(3df,3pd), and M06-2X/aug-cc-pVTZ levels. Also, at the M06-2X/6-311++G(2df,2pd) level, internal reaction coordinate (IRC) analyses were conducted to verify the connection from the transition states to the corresponding products (or reactants). All calculations regarding geometries and frequency were conducted with the aid of the Gaussian 09 (Frisch et al., 2009) program. Furthermore, single-point energies were performed at the CCSD(T)-F12/cc-pVDZ-F12 (Kendall et al., 1992; Adler et al., 2007) level utilizing the ORCA (Neese, 2012) program, based on the optimized geometries mentioned above. The CCSD(T)/aug-cc-pVDZ method was chosen to calculate the relative energies as the fact that, compared with unsigned error (Table S1 in the Supplement) calculated at the CCSD(T)/CBS//M06-2X/6-311++G(2df,2pd) level, unsigned errors calculated at CCSD(T)-F12/cc-pVDZ-F12//M06-2X/6-311++G(2df,2pd) were 0.71 kcal mol⁻¹.

The rate coefficients for the hydrolysis of HNSO₂ with MSA were calculated through a two-step process. Initially, the high-pressure-limit (HPL) rate coefficients were computed applying VRC-VTST methods within the Polyrate package (Chuang et al., 1999). It is worth noting that the electronic structure method for VRC-TST calculations is based on the Gaussian 09 program using M06-2X/6-311++G(2df,2pd). Meanwhile, two pivot points were selected (Bao et al., 2016; Long et al., 2021; Georgievskii and Klippenstein, 2003; Meana-Pañeda et al., 2024) to produce a single-faceted dividing surface for the HNSO₂ hydrolysis (shown in Sect. S1 in the Supplement). Subsequently, on the basis of the HPL rate coefficients, the rate coefficients for the hydrolysis of HNSO₂ with MSA were calculated within the temperature range of 212.6–320.0 K and pressures applying the Master Equation Solver for Multi-Energy well Reactions (MESMER) program (Glowacki et al., 2012). The rate coefficients for the barrierless steps transitioning between reactants and pre-reactive complexes were assessed applying the inverse Laplace transform (ILT) method within MESMER calculations, while the step-transitioning between pre-reactive complexes and post-reactive complexes via transition states was evaluated using the RRKM theory (Mai et al., 2018) in combination with the asymmetric Eckart model. The details of the rate coefficient for the hydrolysis of HNSO₂ with and without X (X = H₂O and MSA) are given in Sect. S1 and Tables 1 and S4.

BOMD simulations were conducted applying DFT implemented in the CP2K program (VandeVondele et al., 2005; Hutter et al., 2014). The exchange and correlation interactions were addressed using the Becke–Lee–Yang–Parr (BLYP) functional (Becke, 1988; Lee et al., 1988), while Grimme's dispersion was applied to address weak dispersion interaction (Grimme et al., 2010). The Goedecker–Teter–Hutter (GTH) conservation pseudopotential (Goedecker et al., 1996; Hartwigsen et al., 1998), combined with a Gaussian DZVP basis set (VandeVondele and Hutter, 2007) and an auxiliary plane wave basis set, was used to represent core and valence electrons. Energy cutoffs (Zhong et al., 2017, 2018, 2019) of 280 Ry for the plane wave basis set and 40 Ry for the Gaussian basis set were applied. The gaseous reactions were simulated in the NVT ensemble at 300 K, with 15×15×15 ų supercells and a time step of 1 fs. To simulate the water microdroplet, the system containing 191 water molecules (Zhong et al., 2017) was utilized in 35×35×35 ų supercells. This setup included HNSO₂ and MSA along with the water drop. Prior to the interfacial simulation, a 10 ps relaxation period in the BOMD simulation was used to equilibrate the water microdroplet system with 191 molecules.

The ACDC model (McGrath et al., 2012; Hu et al., 2023; Zhao et al., 2020; Zhang et al., 2024; Tsona Tchinda et al., 2022; Liu et al., 2021b) was utilized to simulate the (MSA)_x(MA)_y(SFA)_z (0≤y≤x+z≤3) cluster formation rates and explore the potential mechanisms, where the most stable structures of (SFA)_x(MSA)_y(MA)_z (0≤z≤x+y≤3) clusters were searched with ABCluster software (Zhang and Dolg, 2015) (the details are in Sect. S1). This simulation encompasses a variety of temperatures and monomer concentrations to capture the dynamics under different environmental conditions. Thermodynamic parameters, obtained from quantum chemical calculations executed at the M06-2X/6-311++G(2df,2pd) level, were used as inputs for the ACDC model. Notably, many benchmark studies (Zhao et al., 2020; Zhang et al., 2024; Tsona Tchinda et al., 2022; Liu et al., 2021b) show that the M06-2X functional has good performance compared to other common functionals for gaining the Gibbs free energies. For all the M06-2X calculations, the 6-311++G(2df,2pd) basis set was used, as it is a good compromise between accuracy and efficiency and does not yield significant errors in the thermal contribution to the free energy compared to much larger basis sets, such as 6-311++G(3df,3pd), with differences in relative ΔG less than 1.75 kcal mol⁻¹ (Table S7). The temporal progression of cluster concentrations was determined by numerically integrating the birth–death equation, leveraging MATLAB's ode15s solver for enhanced accuracy. 2dcidt=12∑j<iβj,(i-j)cjc(i-j)+∑jγ(i+j)→ici+j-∑jβi,jcicj-12∑j<iγi→jci+Qi-Si Here, ci represents the concentration of a specific cluster, labeled as i; the term βi,j was used to denote the collision coefficient, which was a measure of the frequency at which clusters i and j collide with each other in a given environment or system; the coefficient γ(i+j→i) was defined as the evaporation rate constant that describes the process of a larger cluster, consisting of combined elements i and j, breaking down into the individual smaller clusters i and j; and Qi encompasses all other source terms contributing to the formation of cluster i. Si signifies alternative sink terms for cluster i that may remove it from the system. Considering the formation Gibbs free energy (Table S8) and evaporation rates (Table S9) of all clusters, the clusters containing pure MSA and MA molecules and the clusters containing an SFA molecule are mostly more stable and therefore are allowed to form larger clusters and contribute to particle formation rates. In this case, clusters (MSA)₄ ⋅ (MA)₃, (MSA)₄ ⋅ (MA)₄ and SFA,⋅ (MSA)₃ ⋅ (MA)₃ are set as the boundary clusters.

Given the low chance of three molecules of HNSO₂, H₂O, and MSA colliding simultaneously under atmospheric conditions, the hydrolysis of HNSO₂ with MSA (Channel MSA) was likely a sequential bimolecular process. As the concentration of water molecules (10¹⁸ molec. cm⁻³; Anglada et al., 2013) in the atmosphere is much higher than that of HNSO₂ and MSA (10⁵–10⁹ molec. cm⁻³; Shen et al., 2020), it is hard for the reaction pathway of HNSO₂ … MSA + H₂O to occur in actual atmospheric conditions. Therefore, Channel MSA proceeds through the initial formation of dimers (HNSO₂ … H₂O and MSA. … H₂O) via collisions between HNSO₂ (or MSA) and H₂O. Subsequently, the generated dimer interacts with the third reactant, either MSA or HNSO₂. As seen in Fig. 1, the calculated Gibbs free energy of the MSA … H₂O complex was -0.9 kcal mol⁻¹, which was 4.5 kcal mol⁻¹ lower than that of HNSO₂ … H₂O. Consequently, it was predicted that the primary route for the hydrolysis reaction of HNSO₂ with MSA takes place via the HNSO₂ + MSA … H₂O reaction.

Starting from the HNSO₂ + MSA … H₂O reactants, Channel MSA was initiated through the intermediate complex designated as IM_MSA1. From a geometric perspective, the IM_MSA1 complex exhibits a cage-like configuration by a van der Waals (vdW) force (S1 … O₁, 2.00 Å) and the involvement of three hydrogen bonds: H₂ … O₄ (1.53 Å), H₄ … N₁ (1.60 Å), and H₅ … O₃ (2.07 Å). The Gibbs free energy of the IM_MSA1 complex relative to HNSO₂ + MSA … H₂O reactants was 1.7 kcal mol⁻¹. Subsequently, as presented in Fig. 1, Channel MSA progresses through the transition state TS_MSA1 to yield complex IMF_MSA1. At TS_MSA1, the MSA moiety facilitates two hydrogen atom transfers, with TS_MSA1 lying only 0.8 kcal mol⁻¹ above complex IM_MSA1. Complex IMF_MSA1 exhibits a cage-like structure with a Gibbs free energy 23.4 kcal mol⁻¹ lower than that of IM_MSA1, revealing thermodynamic favorability of HNSO₂ hydrolysis with MSA. To evaluate the relative catalytic impact of MSA and H₂O, Fig. S4 illustrates the profiles of Gibbs free energy for the hydrolysis of HNSO₂ and the corresponding reaction assisted by H₂O. Compared to complex HNSO₂ … (H₂O)₂, the Gibbs stabilization energy of IM_MSA1 increased by 5.6 kcal mol⁻¹, potentially shortening the S₁ … O₁ bond distance by 0.21 Å. Considering the Gibbs free energy barrier and rate coefficients, MSA demonstrates a greater catalytic role compared to H₂O in lowering the energy barrier for the hydrolysis of HNSO₂. In particular, MSA facilitates the hydrogen atom to extraction from H₂O, further reducing the reaction energy barriers to 7.7 kcal mol⁻¹. Meanwhile, the calculated rate coefficients for HNSO₂ hydrolysis with MSA were 3.08×10-11–3.50×10-11 cm³ molec.⁻¹ s⁻¹ within 212.6–320.0 K, exceeding corresponding values for reactions involving H₂O by 2 orders of magnitude. Besides, the Gibbs free energy of IMF_MSA1 was 2.0 kcal mol⁻¹ lower than that of the product complex IMF_WM1 (SFA … H₂O), suggesting SFA has a higher affinity for MSA compared to H₂O. Besides, MSA-assisted HNSO₂ hydrolysis is reduced by 4.9 kcal mol⁻¹ in the energy barrier, as opposed to the NH₃-assisted ammonolysis of SO₃, with its rate constant at 298 K (2.85×10-11 cm³ molec.⁻¹ s⁻¹) close to the value of ammonolysis of SO₃ with NH₃ (4.35×10-10 cm³ molec.⁻¹ s⁻¹) (Li et al., 2018). However, due to the absence of the concentration of HNSO₂, the competitiveness of these two reactions cannot be further confirmed.

To evaluate the comparative catalytic ability of X (X = H₂O and MSA) in the atmosphere, the effective rate coefficients (k′) for X-assisted HNSO₂ hydrolysis were calculated in Table 1. Notably, k′ serves as a metric for gauging the comparative catalytic ability of a series of gaseous catalysts in atmospheric reactions (Sarkar et al., 2017; Zhang et al., 2019, 2020; Buszek et al., 2012; Gonzalez et al., 2011; Parandaman et al., 2018; Anglada et al., 2013). When X was present, the calculated k′ was given by Eq. (3). 3k′X=kX×KeqX…H2O×[X] In Eq. (3), kX was the rate coefficient for X-assisted HNSO₂ hydrolysis (Table 1), while Keq (X … H₂O) denotes the equilibrium coefficients of X … H₂O (Table S2). [X] represents the available concentrations of H₂O (Anglada et al., 2013) and MSA (Shen et al., 2020). As indicated in Table 1, at experimental concentrations ([H₂O] = 5.16×1016–2.35×1018 molec. cm⁻³) within 280.0–320.0 K (at 0 km), the computed kWM′ ranged from 5.99×10-18–7.79×10-17 cm³ molec.⁻¹ s⁻¹. This range exceeded kMSA′ (4.60×10-21–4.81×10-20 cm³ molec.⁻¹ s⁻¹) by 2–4 orders of magnitude, highlighting the pronounced impact of H₂O compared to MSA at 0 km in enhancing the rate of HNSO₂ hydrolysis. However, with the significant decrease in atmospheric water molecules with increasing altitude, the calculated kMSA′ ranged from 1.96×10-19 s⁻¹–1.30×10-17 cm³ molec.⁻¹ s⁻¹, surpassing kWM′ (9.85×10-27–6.51×10-22 cm³ molec.⁻¹ s⁻¹) by 3–10 orders of magnitude. This illustrates that MSA has a significantly greater catalytic ability than H₂O in accelerating the rate of HNSO₂ hydrolysis within 5–15 km. So, HNSO₂ hydrolysis with MSA may represent a potential formation pathway for SFA across an altitude scope of 5–15 km.

The interfacial mechanism of MSA-assisted HNSO₂ hydrolysis at the air–water interface has not been thoroughly investigated. Interestingly, our simulations show that HNSO₂ and MSA molecules spend approximately 49.1 % and 12.1 % of the time, respectively, at the air–water interface during the 150 ns simulation (Figs. S1 and S6). This reveals that the presence of HNSO₂ and MSA at the air–water interface should not be disregarded. Therefore, BOMD simulations were performed to clarify the interfacial mechanism of MSA-assisted HNSO₂ hydrolysis at the air–water interface. Comparable to the reactions of SO₃ at the air–water interface with acidic molecules (Cheng et al., 2023; Zhong et al., 2019), the hydrolysis of HNSO₂ with MSA at the air–water interface may occur through three pathways: (i) the adsorbed MSA interacts with HNSO₂ at the air–water interface, (ii) the adsorbed HNSO₂ interacts with MSA at the air–water interface, and (iii) the HNSO₂ … MSA complex reacts at the air–water interface. Nevertheless, because of the high reactiveness of MSA at the air–water interface, the lifetime of MSA was minimal (seen in Fig. S9) on the water droplet, which was around a small number of picoseconds, leading to the rapid formation of MSA⁻ ion. Meanwhile, although HNSO₂ remains stable at the air–water interface (seen in Fig. S8) and does not dissociate within 10 ps, the hydrated form of HNSO₂ illustrated in Fig. S8 was not conducive to HNSO₂ hydrolysis at the air–water interface. Therefore, model (iii) was primarily considered for HNSO₂ hydrolysis with MSA at the air–water interface. It is worth noting that the HNSO₂ … MSA complex can persist at the air–water interface for approximately 34.2 % of the 150 ns simulation time (see in Fig. S7). For model (iii), two types of reactions were found at the air–water interface: (a) the NH₂SO3- and H₃O⁺ ion formation mechanism and (b) the proton exchange mechanism.

NH₂SO3- and H₃O⁺ ion formation mechanism. Figures 2a and S10 and Movie S1 illustrate the formation mechanism of NH₂SO3- and H₃O⁺ ions through the chain structure at 4.57 ps, and a chain-hydrolyzed transition state was observed, accompanied by two proton transfer events. IN particular, an H₂ atom transferred from the OH moiety of an MSA molecule to the terminal N atom of an HNSO₂ molecule, resulting in the breaking of the O₃–H₂ bond (with a length of 1.49 Å) and the formation of an H₂–N bond (with a length of 1.14 Å). Concurrently, an interfacial water molecule decomposes, leading to the elongation of the O₁–H₁ bond to over 1.00 Å, with the S1 atom of HNSO₂ obtaining the OH moiety of the interfacial water molecule (d(S1–O1) = 1.60 Å). By 4.61 ps, the N–H₂ and S₁–O₁ bonds both shortened to 0.99 and 1.01 Å, revealing the formation of the SFA molecule. However, due to its strong acidity, the SFA molecule could only persist on the water droplet surface for a picosecond timescale. As a result, at 7.43 ps, the proton of SFA transferred to another interfacial water molecule, completing the deprotonation of SFA. The loop structure mechanism (Figs. 2b and S11 and Movie S2) was similar to the chain structure mechanism. However, in this case, the proton of SFA transferred to CH₃SO3- rather than to an interfacial water molecule.

Proton exchange mechanism. As depicted in Fig. 3, the proton exchange mechanism illustrates the deprotonation of MSA concurrent with HNSO₂ hydration at the air–water interface. As shown in Figs. 3a and S12 and Movie S3, MSA-mediated HNSO₂ hydration with a single water molecule was observed. Initially, the HNSO₂ … MSA complex quickly associates with an interfacial water molecule and forms a loop structure complex that accelerates the rate of proton transfer. By 4.38 ps, an eight-membered loop structure complex, HNSO₂ … H₂O … MSA, emerges, characterized by two hydrogen bonds (d(H2–N)=1.82 Å and d(H1–O2)=1.92 Å) and a van der Waals force (d(S1–O1)=2.35 Å). Thereafter, at 4.77 ps, a transition-state-like configuration was identified where the water molecule within the loop complex dissociated, elongating the O₁–H₁ bond to over 1.00 Å, and the S atom of HNSO₂ attached to the OH group of the interfacial water molecule. Concurrently, the CH₃SO3- ion receives the proton from the separated interfacial water molecule. The entire reaction for MSA-mediated HNSO₂ hydration with one water molecule was completed at 4.80 ps, resulting in the formation of SFA and MSA molecules. MSA-mediated hydration of HNSO₂ with two water molecules (Figs. 3b and S13 and Movie S4) at the air–water interface was similar to the mechanism identified with one water molecule. However, the inclusion of two water molecules enlarges the loop, significantly reducing the stress on the loop structures. Consistent with the prediction in Fig. 4, the loop structures preferred to include two water molecules rather than one water molecule. This observation agrees well with the reported hydration of the Criegee intermediate at the air–water interface (Zhu et al., 2016; Kumar et al., 2018; Liu et al., 2021a; Zhang et al., 2023a). Additionally, MSA-mediated hydration of HNSO₂ with three water molecules (Fig. S14 and Movie S5) was observed in the proton exchange mechanism. However, its probability of occurrence was smaller due to the relatively larger entropy effect. It was noteworthy that the SFA and MSA molecules formed in the proton exchange mechanism were not stable at the air–water interface, which can further interact with an interfacial water molecule to form the corresponding ions of NH₂SO3- and CH₃SO3-.

At the air–water interface, a sum of 50 BOMD trajectories, each lasting 10 ps, was conducted to investigate HNSO₂ hydrolysis with MSA. Two distinct mechanisms were observed: the formation of NH₂SO3- and H₃O⁺ ions (shown in blue and yellow in Fig. 4) and the proton exchange mechanism (represented by orange, purple, and green in Fig. 4). In the mechanism involving the formation of NH₂SO3- and H₃O⁺ ions, approximately 22 % (Figs. 2a, 4, and S10 and Movie S1) of the reactions took place via a chain structure, while the majority (∼18 %) (Figs. 2b, 4, and S11 and Movie S2) proceeded through a loop structure mechanism. This discrepancy can be attributed to the uncertainty regarding the direction of proton transfer from SFA. Since the number of water molecules near the water microdroplet far exceeded that of CH₃SO3-, protons were predominantly transferred to interface water molecules, making the loop structure mechanism weaker than the chain structure mechanism. Approximately 60 % of the reactions were observed to be due to the proton exchange mechanism in BOMD simulations. Through water-mediated mechanisms, these reactions resulted in SFA formation. Similarly to gas-phase reactions, loop structures were observed in these reactions. Approximately 10 % of the reactions formed a loop structure involving one water molecule (Figs. 3a, 4, and S12 and Movie S3), while the most common loop structure involved two water molecules (about 42 %) (Figs. 3b, 4, and S13 and Movie S4). Smaller loops were found to experience more stress than loop structures with two water molecules. In cases of loop structures with three water molecules (about 8 %) (Figs. 4 and S14 and Movie S5), the entropy effect was deemed to be more significant than the strain effect and likely played a dominant role. The two water molecules contained in the loop structure not only acted as a reactant but also facilitated proton transfer as a bridge.

As the discussion above, the formation of SFA⁻ … H₃O⁺ and MSA⁻ … H₃O⁺ ion pairs can occur within a few picoseconds at the air–water interface. The atmospheric affinity of MSA⁻, SFA⁻, and H₃O⁺ for gaseous precursors was further probed by evaluating the free energies of interaction. It was worth noting that compounds such as MSA, MA, HNO₃ (NA), and (COOH)₂ (OA) were identified as candidate species for consideration (Wang et al., 2024; Kulmala et al., 2004). As presented in Table 2, the computed binding energies demonstrate that the interactions of SFA⁻ … MSA, SFA⁻ … NA, SFA⁻ … OA, H₃O⁺ … MA, MSA⁻ … MSA, MSA⁻ … OA, and MSA⁻ … NA were stronger than those of MSA … MA (one of the primary precursors for atmospheric aerosols), with their Gibbs free energies increased by 14.3–50.9 kcal mol⁻¹. The findings indicate that the presence of SFA⁻, MSA⁻, and H₃O⁺ at the interface facilitates the capture of potential gaseous species onto the surface of the water microdroplet.

Furthermore, we investigated the possibility of SFA⁻ contributing to the enlargement of particles within the MSA-MA cluster, taking into account the geometric configuration and the free energy of formation for the (MSA)₁ ⋅ (MA)₁ ⋅ (SFA-)1 clusters aggregating. Compared with other clusters, such as (MSA)₁ ⋅ (MA)₁ ⋅ (X)₁ (where X = HCOOH, CH₃COOH, CHOCOOH, OA, CH₃COCOOH, HOOCCH₂COOH, HOOC(CH)₂COOH, HOOC(CH2)2COOH, HOOC(CH2)3COOH, C₆H₅(COOH), and C₁₀H₁₆O₃) clusters (Zhang et al., 2022a), the quantity of hydrogen bonds within the (MSA)₁ ⋅ (MA)₁ ⋅ (SFA⁻)₁ cluster increased, and the loop of complex was expanded. It has been demonstrated that SFA⁻ has the greatest capacity to stabilize MSA-MA clusters and facilitate MSA-MA nucleation in these clusters. This was attributed to its acidic nature and structural characteristics, which include a greater number of intermolecular hydrogen bond binding sites. Therefore, relative to the (MSA)₁ ⋅ (MA)₁ ⋅ (X)₁ cluster (Table 2), the Gibbs formation free energy ΔG of the (MSA)₁ ⋅ (MA)₁ ⋅ (SFA⁻)₁ cluster was lower, indicating that the NH₂SO3- ion exhibits a more potent nucleation capacity at the air–water interface compared to the X species in the gas phase. Consequently, our forecast was that the presence of NH₂SO3- at the air–water interface would foster enhanced particle growth.