| 30 Sep 2025
Effect of humidity on the first steps of atmospheric new particles formation: Computational study of hydrated molecular clusters
Abstract. To improve computational modeling of hydrated atmospheric molecular clusters, we systematically evaluated quantum-chemical methods for predicting accurate structural and energetic properties of clusters containing a variety of atmospherically relevant acids and bases, with up to five water molecules. We find that the commonly applied ωB97X-D/6-31++G(d,p) method with DLPNONormalPNO–CCSD(T0)/aug-cc-pVTZ electronic energy correction is suitable for hydrated clusters. Composite density functional methods such as B97-3c, r2SCAN-3c and ωB97X-3c are effective for pre-screening or modeling large clusters, while the local natural orbital approach LNO–CCSD(T)/aug′-cc-pVTZ is well-suited for accurate refinement due to its low memory requirements, high accuracy, and favorable computational scaling. Nevertheless, the ωB97X-3c method has a reasonable accuracy even without the electronic energy correction.
We also assessed thermochemical corrections beyond the conventional harmonic oscillator approximation applied only to the lowest free-energy structure. For the limiting cases of no corrections and the ideal maximum corrections, we calculated hydration distributions and particle formation rates, with a specific emphasis on sulfuric acid–ammonia (SA–AM), sulfuric acid–dimethylamine (SA–DMA), and methanesulfonic acid–methylamine (MSA–MA) clusters. Hydration of small clusters is generally limited, with only selected SA- and MSA-containing clusters showing substantial hydration. Due to the high water concentration in the atmosphere, hydration equilibrates fast, increasing the number of accessible states, and thus stabilizing clusters. However, its effect on cluster formation and new particle formation is highly system dependent.
MSA–MA particle formation rates are more sensitive to hydration than those of SA–AM or SA–DMA, though the enhancement remains modest. Despite being more hydrated than SA–DMA clusters, MSA–MA clusters form new particles at relatively low rates, comparable to SA–AM. Under typical atmospheric conditions, SA–DMA is expected to dominate new particle formation, even at high humidity.
Competing interests: At least one of the (co-)authors is a member of the editorial board of Aerosol Research. The authors have no other competing interests to declare.
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The authors conducted a systematic study of cluster hydration in atmospherically important systems. This purely theoretical study primarily focuses on quantum chemical approaches and evaluates them against one another. Such benchmarking is standard yet crucial for future studies. Additionally, the authors enhance their investigations by exploring cluster properties beyond the standard quantum chemical framework. I commend the authors for their efforts in this regard!
The benchmarking is extensive and done with high rigor. The group’s ability of carrying out such method comparisons is well known and overall top quality, and I have very little to comment or criticize about.
A general remark: While the authors note that no good reference data exist for cluster thermodynamics, I find it useful and interesting that they investigate the potential effects of anharmonicities. These systems likely exhibit significant thermal fluctuations, making the standard harmonic approach inadequate. Therefore, I believe the results and speculation are sufficient as they are and are generally highly welcome.
The nucleation rate section (3.5) is clearly the weakest part of the study, which is unfortunate because it discusses the real-life implications. I will elaborate on my critique below:
In the ACDC simulations, the limiting size is set to 3 acids, 3 bases, and any number of water molecules. Given the studied concentrations, the system size is too small. Except for a few extreme cases, the critical cluster is not included in the set of studied clusters. This is briefly mentioned on Line 465, but not very clearly. Additionally, in the supplementary information, they state:
“Because the maximum simulated cluster size was relatively small, the critical cluster size, where growth starts to vastly outweigh evaporation, might not be included” and “Elm et al.(6) coined the term: potential particle formation rate Jpotential, to indicate that the results cannot directly be related to the actual particle formation rate J, but are rather a measure for the importance of different compounds in cluster formation.”
And the article cited in the SI (Elm et al., 2017) does not include the concept or its coining. It is quite frustrating that authors miscite, but come on, it's one of your own!
If nothing else, I strongly think the authors should at least explicitly state that the nucleation rates studied are not true nucleation rates. Currently, the results may mislead readers into thinking otherwise. As noted in the SI, the measure used here is the potential particle formation rate, which could serve as a good proxy. However, given the magnitude of the effect of hydration (within one order of magnitude), this analysis needs to be more rigorous.
Let me elaborate on my point: The idea that incomplete cluster formation free energies can indicate the magnitude of the nucleation rate applies when comparing nucleation capabilities among different chemistries (e.g., SA-AM to SA-DMA). However, in this case, the comparison is made against hydration or, more precisely, relative humidity. The authors demonstrate, particularly in fig. 7 and 8, that the level of hydration depends on cluster size in a non-monotonic manner. Therefore, extrapolating the level of hydration to larger, unknown clusters lacks proper justification.
To provide a more theoretical perspective: The nucleation rate largely depends exponentially on the free energy of the critical cluster only: J ~ exp(-dG*/RT). Thus, J primarily depends on the height of the nucleation barrier, not its slope. Consequently, if hydration significantly decreases free energy at the critical size but has smaller effect around it, the nucleation rate will increase notably and “unpredictably”. The same reasoning applies in reverse if the critical cluster is hydrophobic among otherwise hydrophilic clusters.
I really don’t understand why the authors chose to study these concentrations. Since they are not directly comparing their results to any experimental or measurement data, they could select conditions where the critical cluster appears within the set of clusters. I believe that limiting the range of critical clusters is a lesser issue than simulating incomplete systems. I kindly suggest that they consider running the simulations at other concentrations.
Some minor comments and questions:
Line 207: Coagulation loss is referred to as CL, but it is also CS (SI, p. S6). Should CL/CS have a unit?
Line 306: The authors state that they have considered multi-conformer Boltzmann averaging, as explained later in Section 3.3.3. However, eq 4 does not resemble Boltzmann averaging. Could the authors elaborate on this approach?
Figure 8 is somewhat difficult to read because of the various colors, markers, line types, and shaded areas. I suggest that the authors consider dividing the data into three separate plots or finding an alternative way to present it for maximum clarity. Otherwise, the visual elements of the manuscript are clear and well presented.
L460: “Since evaporation rates depend exponentially on binding free energies, an error of just 3 kcal mol−1 can likewise produce a factor of 20 difference in J.” Is this correct? I would assume it is 3RT, approximately 1.8 kcal/mol: exp(1.8 kcal/mol / RT) = exp(3) = 20.