The Ionization Energies of Dust-Forming Metal Oxide Clusters
Round 1
Reviewer 1 Report
Authors are presenting a DFT based computations of the ionization energies of clusters of magnesia (MgO)n, silicon monoxide (SiO)n, alumina (Al2O3)n
7 and titania (TiO2)n with stoichiometric sizes of n=1 - 8. The computations are standard and there are too many drawbacks with this work:
1) What is the nature of the bounding? Long range interactions are not considered. Dispersion effects are important for thee systems.
2) Authors are considering the species in their global minimum. Why? The temprature of the media is far from 0K to validate this asumption. Indeed, nothing is said about the orther less stable stable that can also be formed at T of the ABG.
3) Some ionization energies are 1 eV off. This is too much! Thus DFT is not suited for the derivation of such quantities.
4) How the initial structures of the ions were chosen? One may expect different ionic and neutral forms.
5) Important references are missing:
e.g for MgO clusters
Phys. Chem. Chem. Phys., 2019,21, 23102-23110
Phys. Chem. Chem. Phys., 2001,3, 5024-5034
J. Phys. Chem. A 2020, 124, 1, 101–107
etc.
6) No MO analysis supprots the electron removal induced changes. Only speculations.
7) I am not sure that authors did find the most stable ionic forms. This is misleading for non experts.
8) Again, nothing is said about the composition of the wavefunctions of the species under investigation. Are they monoconfigurational to allow the use of DFT? This is really questionnable for all species and especially those containing Ti.
9) Not much comparison t experiments for validation.
Author Response
After a thoughtful and thorough inspection, we decided to reject this referee report by reviewer #1. Reviewer #1 generally doubts the validity of the methods and principles (i. e. density functional theory (DFT) calculations) that we use to derive our results. These concerns on the validity of DFT are expressed in points 1-3, 6 and 8.
DFT methods are widely used in the chemical and astrophysical community and have been proven to accurately predict potential energies of molecular systems at affordable computational costs (see e.g. van Mourik, T. Buehl, M. and Gaigeot M., 2014, https://doi.org/10.1098/rsta.2012.0488).
Moreover, reviewer #1 does not suggest an alternative to DFT calculations. The only alternative, that comes to our mind, are high level-of-theory post-Hartree-Fock calculations. These calculations are expensive and beyond the scope of this paper. Moreover, they are even prohibitive for large cluster sizes.
As a follow-up paper of Boulangier et al (2019), it would also be at odds to resort to a different methodology, making a comparison of the neutral and ionized clusters practically impossible.
In point 5, reviewer #1 suggests to include additional literature on MgO clusters.
However, the suggested papers do not contain new lowest-energy (MgO)n cluster structures and hence, they do not add any extra value to our paper or change its conclusions.
Generally, the report uses a clunky phrasing. Some sentences don’t even include a verb.
Reviewer 2 Report
See attached file
Comments for author File: 
Comments.pdf
Author Response
We thank the referee for his reasonable comments. In particular, we appreciate her/his helpful comments on ion-molecule chemistry and photo-ionization. In the following, we will address the comments in detail and indicate what changes have been made in the manuscript.
1. “It should be better explained why the particular four kinds of cluster series are considered as to be of possible relevance for nucleation. The discussion in the introduction remains somewhat vague, in particular since there is already some literature on this point (and this should be cited!).”
This study represents a continuation of a prior work by Boulangier et al. (2019) who investigated the neutral clusters of these four nucleation species. Boulangier et al 2019 discussed in detail, why these nucleation candidates were chosen. Therefore, we emphasize by adding to the Introduction:
This study represents a continuation of [15] who investigated the nucleation of neutral clusters of four metal oxide families (MgO, SiO, Al2O3, TiO2). Here, we address the corresponding ionization energies and its related cations allowing us to assess whether a fast ion-molecule chemistry can compete with a comparatively slow neutral-neutral nucleation.
Moreover, we explicitly mention previous theoretical and experimental studies that investigated the cluster nucleation of the four considered species in an astronomical context.
Previous studies have addressed the four nucleation candidates MgO, SiO, Al2O3, and TiO2, respectively, and described the properties of the related neutral clusters in astrophysical environments theoretically [19–23], as well as experimentally [24–26].
2. “The link "http://dave202.bplaced.net/ions/" to the data obtained in this paper at the end of the paper does not work. Without this it is difficult to judge how useful the results obtained in this paper are for the astronomical scientific community. Please make sure that the data can be assessed.”
We agree with the referee that it is essential in Astronomy and any other science field to provide the scientific community with accessible, relevant data.
The reason that this publicly available link was not working yet is that we aim to protect ourselves against any sort of plagiarism; in particular, as the article manuscript has not (yet) been accepted.
Therefore, we uploaded some cation structures to show that the given URL is working and can send the referee confidential e-mail, if she/he wishes so. Once our paper is finally accepted, we will provide a complete data set at http://dave202.bplaced.net/ions/
3. “The paragraph from line 272 to line 287 probably aims to demonstrate that ionized clusters could exist in chemical equilibrium in regions of enhanced temperatures, e.g. behind shocks, and the ionized clusters then could be important for the chemistry of dust formation. But no results of this or of any other kind of result are given. The whole paragraph, as it stands, makes no sense because it remains unclear what is actually demonstrated. This paragraph should be reformulated to make clear what the authors wish to demonstrate.”
It is a valid point of the referee to question the assumption of chemical equilibrium in shocked gas layers, which is somewhat misleading / contradictory. This paragraphs aims to show that even if we take into account the effect of recurrent shocks depositing thermal energy in the gas, the ionization fraction will be small. The presence of a radiation field (photons) is more likely to be the source of ionization than energetic gas-phase collisions. We restructured this paragraph (and the subsequent paragraph mentioned by the referee in point 4) and gave some estimates for the ionization fraction for characteristic AGB temperatures.
“Gas constituents have typically not a single temperature, but follow a
statistical distribution function. Assuming chemical equilibrium, we can asses the fraction of ionized clusters by using the Saha equation:
where n(X+) and n ( X ) are the number density of the ionized and neutral cluster
species X, respectively. E i is the previously introduced ionization energy and KT correspond to the thermal energy. The factor 2/1 arises from the spin multiplicity M of the neutrals (M=1) and cations (M=2) and accounts for the statistical weights. Note that for the triplet X=Al2O3, it should read 2/3.
From Table 1 it is apparent that the ionization energies range from 6.62 eV to 11.49 eV corresponding to equivalent kinetic temperatures of 76 800 − 133 300 K. These temperatures are orders of magnitude larger than the prevailing photospheric temperatures of an AGB star, ranging typically between 2000 K and 3000 K.
According to equation (2) the ionization fractions at the photosphere result in values between 2.3e-29 and 1.5e-11 foR t=2000-3000 k.
Pulsation-induced shocks, travelling periodically through the circumstellar envelopes of AGB stars, increase the temperature locally and temporarily.
By applying the Rakine-Hugoniot jump conditions to photospheric conditions and a diatomic gas, temperatures > 40000 K can be attained (see e.g. [34]). This is still a factor of 2 − 3 smaller than the equivalent ionization temperatures.
However, owing to its intrinsic chemical equilibrium assumption, equation (2) is not applicable to an immediate post-shock gas strongly deviating from equilibrium conditions.”
4. The paragraph from line 288 to line 306 argues that UV radiation leaking through the pores of a clumpy circumstellar shell or from shocked gas can ionize the clusters of the type considered in the paper and may drive an ion-molecule chemistry. Also here it is unclear, what really is demonstrated in this paragraph. The possibility that UV fields may play a significant role for the chemistry is evident. The consideration on the UV radiation from a hot gas is trivial and the conclusion from this is even not correct, because it is not necessary that temperatures of 15500 -27000 K are required to ionize the clusters. This would already happen for a small fraction of clusters for lower temperatures. The essential question is how low the fraction of ions can be that by virtue of the orders-of-magnitude higher reaction rates of ion-molecule reactions compared to neutral-neutral reactions the ion-molecule chemistry may compete with or dominate over the chemistry of neutral-neutral reactions. This should be discussed more precisely. Alternatively one cake recourse to the apparently only two papers that have discussed in some detail the role of ion-molecule chemistry for red giants up to now (Glassgold & Huggins ApJ 306, 605 and Beck et al. A&A 265, 626)
Formula 2 in the previous paragraph implies a Boltzmann distribution of the cations and thus a small fraction of the clusters located in the high energy tail of the distribution exist indeed in an ionized state (also for low temperatures). We agree with the referee’s argument and included an estimation of the species flux (i.e. rate (typical values in the KIDA database) multiplied by concentration) by stating
Although the ionization fraction is comparatively low at characteristic AGB temperatures, the orders-of-magnitude faster ion-molecule chemistry with rates up to 100 cm3/s could compete with neutral-neutral reaction rates that usually range between 10-9 – 10-13 cm3/s.
Accounting for both, the fast kinetic rates (i.e. 9-13 order of magnitude) and the low abundances of the cluster cations (i.e. 11-29 order of magnitude), we find that the ion-molecule chemistry can compete with neutral-neutral reactions for the lowest ionization energies (< 8 eV) and the fastest ion-molecule rates. Moreover, pulsation-induced shocks are prone to increase the amount of ions and therefore also enhance their impact on the circumstellar chemistry. Therefore, an ion-molecule driven chemistry can become import in the dust formation zone of AGB stars, in particular in the immediate post-shock regime.
However, neutral-neutral reactions are still believed to represent the dominant nucleation mechanism, as only extreme cases (i.e. reactions with the fastest ion-molecule rates and the largest ion concentrations corresponding to low ionization energies) occur on comparable timescales.
Reviewer 3 Report
Grain formation around AGB stars is crucial for understanding the driving mechanism of the AGB mass-loss wind, an important phenomenon in stellar evolution and chemical enrichment of the universe. However, due to the lack of knowledge about many physical quantities, the grain formation process is still poorly understood. In this paper, one such unknown parameter is reported: the ionization energy of molecular clusters that are expected to be formed in the early stages of the grain formation process. This could be important fundamental knowledge for future studies of the grain formation process. Therefore, I would like to recommend the publication of this paper with some revisions. The points that should be corrected are listed below.
L.7: Insert a comma between "alumina" and "and".
L.26-27: The condensation temperature depends on the environment, and the described values are appropriate for the circumstellar environment around AGB stars. Therefore, I would like to recommend inserting an expression such as "around AGB stars" after "900-1300 K".
L.27: I would recommend mentioning the specific size of "large circumstellar dust grains".
L.47-48: L.47-48: I would like to recommend mentioning the possibility of ultraviolet radiation from hot companion stars like "Mira B".
L.104-106: The authors describe that for all four cluster families, the minimum of the vertical ionization energy is found at the largest cluster size (n=8). However, looking at Figure 1, the minimum vertical ionization energy of TiO2 cluster is found at n=7. Therefore, I would like to request the authors to reconfirm the consistency of this statement with their results.
L.262: I would like to ask the author to include the specific value of the ionization energy of Mg.
L.277: Rakine -> Rankine
L.278: It would be better to use the "∼" mark instead of the ">" mark. The ">" mark would include the higher temperature and make the comparison described in the next line difficult.
L.281: asses -> assess
L.281-287: The Saha equation is given, but the results of the evaluation of the ionization degree using this equation are not given. I would like to ask the authors to describe the evaluated ionization degree and comment on whether it is negligible or not.
L.282: In my understanding, Saha's equation includes terms such as electron number density and (2 π me kT)1.5/h3. However, the equation in the manuscript does not include them and looks like a simple Boltzmann equation. If this equation is derived under some assumption, I would like to request the authors to describe that assumption.
L.296: I think that the Wien displacement law is valid in the lower energy/temperature regime and that the declaration in L.296 is not necessary. If so, I would like to suggest removing this sentence.
L.300-301: I understand that radiation with a temperature of 15500-27000 K is efficient to ionize the clusters. However, whether or not such a radiation source is available is important for this discussion. I would like to ask the authors to explain the possibility of the radiation sources they envision.
L.308: I think that maser radiation itself does not contribute to ionization, because its photon energy is too low. Therefore, I could not understand the sentence "pumped maser emission...clusters". I would like to recheck this point and correct it if necessary.
L.311: ionzie -> ionize
Author Response
We thank the referee for her/his time and her/his valuable comments. We agree with all points and have addressed and implemented them accordingly in the manuscript (highlighted in bold font). In the following, we address the comments in detail:
-L.7: Insert a comma between "alumina" and "and".
We adapted the manuscript accordingly.
-L.26-27: The condensation temperature depends on the environment, and the described values are appropriate for the circumstellar environment around AGB stars. Therefore, I would like to recommend inserting an expression such as "around AGB stars" after "900-1300 K".
We adapted the manuscript accordingly.
-L.27: I would recommend mentioning the specific size of "large circumstellar dust grains".
We added a grain size of 0.6 microns.
"However, large circumstellar dust grains with sizes of 0.6 μm form within two stellar radii, ..."
-L.47-48: L.47-48: I would like to recommend mentioning the possibility of ultraviolet radiation from hot companion stars like "Mira B".
This is a good point of the referee. Radiation from a binary companion can indeed aid the ionization of clusters. We included the following sentence:
“In addition, some AGB stars reside in binary systems with a hot companion star providing a source of ionizing UV radiation, as for example Mira B.”
-"L.104-106: The authors describe that for all four cluster families, the minimum of the vertical ionization energy is found at the largest cluster size (n=8). However, looking at Figure 1, the minimum vertical ionization energy of TiO2 cluster is found at n=7. Therefore, I would like to request the authors to reconfirm the consistency of this statement with their results."
We agree with the referee as we mistakenly used the term vertical instead of adiabatic and changed the sentence into:
"With the exception of (MgO)5 , all four cluster families show their lowest adiabatic ionization energy at their largest cluster size (n=8)."
-L.262: I would like to ask the author to include the specific value of the ionization energy of Mg.
We include the atomic Mg ionization energy of 7.65 eV
-L.277: Rakine -> Rankine
corrected.
-L.278: It would be better to use the "∼" mark instead of the ">" mark. The ">" mark would include the higher temperature and make the comparison described in the next line difficult.
We adapted the manuscript accordingly.
-L.281: asses -> assess
We adapted the manuscript accordingly.
-L.281-287: The Saha equation is given, but the results of the evaluation of the ionization degree using this equation are not given. I would like to ask the authors to describe the evaluated ionization degree and comment on whether it is negligible or not.
We gave a range of values of ionization fractions for photospheric conditions.
According to equation 2 the ionization fractions at the photosphere result in small values between 2.3 × 10 −29 and 1.5 × 10 −11 for T = 2000 − 3000 K."
Furthermore, we added in the discussion section we
Though the ionization fraction is comparatively low at characteristic AGB temperatures, the orders-of-magnitude faster ion-molecule chemistry with rates up to 100 cm 3 s − 1 could compete with neutral-neutral reaction rates that usually range between 10-9 − 10−13 cm 3 s − 1 . Accounting for both, the fast kinetic rates (i.e. 9 − 13 orders of magnitude) and the low abundances of the cluster cations (i.e. 11 − 29 orders of magnitude), we find that the ion-molecule chemistry can compete with neutral-neutral reactions for the lowest ionization energies (< 8 eV) and the fastest ion-molecule rates."
-L.282: In my understanding, Saha's equation includes terms such as electron number density and (2 π me kT)1.5/h3. However, the equation in the manuscript does not include them and looks like a simple Boltzmann equation. If this equation is derived under some assumption, I would like to request the authors to describe that assumption.
We agree with the referee that the equation used here is the Boltzmann equation and not the Saha equation taking explicitly into account the electron contribution. Therefore, we replaced "Saha equation" with "Boltzmann law".
-L.296: I think that the Wien displacement law is valid in the lower energy/temperature regime and that the declaration in L.296 is not necessary. If so, I would like to suggest removing this sentence.
Owing to a suggestion from another reviewer we revised our conclusion and the Wien displacement law is not used in the discussion anymore.
-L.300-301: I understand that radiation with a temperature of 15500-27000 K is efficient to ionize the clusters. However, whether or not such a radiation source is available is important for this discussion. I would like to ask the authors to explain the possibility of the radiation sources they envision.
Owing to a suggestion from another reviewer we revised our conclusion. However, we mentioned the possibility that "radiation from a hot companion star can provide the energy required to ionize (part of) the metal oxide clusters."
-L.308: I think that maser radiation itself does not contribute to ionization, because its photon energy is too low. Therefore, I could not understand the sentence "pumped maser emission...clusters". I would like to recheck this point and correct it if necessary.
-We agree with the referee that the photon energies of the masing molecules like e.g. SiO are far below the ionization energies of the clusters and removed the sentence.
-L.311: ionzie -> ionize
corrected
Author Response File: Author Response.pdf
Round 2
Reviewer 2 Report
The paper is now OK, no further comments
