the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
The ion-ion recombination coefficient α: Comparison of temperature- and pressure-dependent parameterisations for the troposphere and lower stratosphere
- Institute for Atmospheric and Environmental Sciences, Goethe University Frankfurt am Main, Frankfurt am Main, 60629, Germany
- Institute for Atmospheric and Environmental Sciences, Goethe University Frankfurt am Main, Frankfurt am Main, 60629, Germany
Abstract. Many different atmospheric, physical and chemical processes are affected by ions. An important sink for atmospheric ions is the reaction and mutual neutralisation of a positive and a negative ion, also called ion-ion recombination. While the value for the ion-ion recombination coefficient α is well known for standard conditions (namely 1.7 · 10–6 cm3 s–1), it needs to be calculated for deviating temperature and pressure conditions, especially for applications in higher altitudes of the atmosphere. In this work, we review the history of theories and parameterisations of the ion-ion recombination coefficient, focussing on the temperature and pressure dependencies and on the altitude range between 0 and 20 km. Starting from theories based on J. J. Thomson’s work, we describe important semi-empirical adjustments as well as field, model and laboratory data sets, followed by a short review of physical theories that take the microscopic processes during recombination into account, including a molecular dynamics approach. We present a comparison between all theories, parameterisations, field, model, and laboratory data sets to conclude on a favourable parameterisation. While many theories agree well with field data above approx. 10 km altitude, the nature of the recombination coefficient is still widely unknown between Earth’s surface and an altitude of 10 km. According to the current state of knowledge, it appears most reasonable to assume a constant value for the recombination coefficient for this region, while we recommend using a parameterisation for altitudes above 10 km. Overall, the parameterisation of Brasseur and Chatel (1983) shows the most convincing results. The need for future research, be it in the laboratory or by means of modelling, is identified.
Marcel Zauner-Wieczorek et al.
Status: final response (author comments only)
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RC1: 'Comment on acp-2021-795', Anonymous Referee #1, 21 Nov 2021
This manuscript reviews and then attempts to apply various theories predicting the ion-ion recombination rate in comparison to select experimental data. I think this is certainly a topic worthy of study. However, my recommendation is between major revision and rejection, because I believe this work is misguided in its approach and has more than several inaccurate statements in it. Much of this has to do with the manner in which the authors compare to “Ta20” (which is really a comparison to Fuchs 1963, not Ta20), and I do not believe the “intercomparison” of theories approach with subjective choices in inputs is a reasonable way to go about scientific study. Ultimately, I would like to think the authors can improve upon this work, and endorse major revision.
Comments:
1. As noted in the prior paragraph, I believe the article is quite misguided in its approach. I think this is most apparent in the works presentation and discussion of Tamadate et al (2020). The article devotes quite a bit of time discussing the work of Tamadate et al (2020), and in fact in looking at Tamadate et al 2020, it looks like a considerable fraction of the review section of this article is based upon the introduction of Tamadate et al. Specifically, a large fraction of the references reviewed in this work are similarly discussed in Tamadate et al, and the symbols and notation in the present manuscript also appear to be taken directly from Tamadate et al in the number of instances (this manuscript even refer readers to Tamadate et al 2020 for Hoppel and Frick’s equations, instead of referring readers to Hoppel and Frick!). However, in reading Tamadate et al (2020), I come away thinking the authors completely miss the purpose of that article (or for some reason, do not want others to consider developing and expanding the Tamadate et al 2020 approach). Tamadate et al (2020), and the same researchers subsequent study (doi: 10.1039/D0CP03989F) focus explicitly on leveraging Molecular Dynamics simulations to determine the collision probability needed for implement in Filippov’s generalized version of the limiting sphere theory. When the authors attempt to compare their prior measurements to Tamadate et al (2020), they state:
“We calculated Ta20 based on the derivations after Fuchs (1963) (Eq. (34) to (36)) and after Filippov (1993) (Eq. (37) to (39)), using Eq. (40) to (42) likewise. However, both derivations yielded the same results within our limits of uncertainty, therefore, for a better overview, for Ta20 we only show the results based on Fuchs.”
Using the relationship of Fuchs 1963 with the limiting sphere of Wright is nothing more than Fuchs original theory, and not a test of what Tamadate et al did (equations 34-36 are Fuchs's exact theory, Tamadate et al just reiterates them). To be clear, Tamadate et al did not derive new equations, they implement Filippov’s equations with Molecular Dynamics simulations, and without using results from MD simulations specific to the ions, gas composition, temperature, and pressure of interest, comparison is not being made appropriate to their work. The simulations in Tamadate et al (2020) agree with Fuchs when they neglect gas molecule-ion interactions (validating their approach), but this is not intended to be an accurate calculation of the recombination rate. Their simulations lead to much higher recombination rates than those of Fuchs and would lead to different values than the predictions here. I suggest correcting this comparison to note it is a comparison to Fuchs’s theory, not to Tamadate et al’s hybrid continuum-MD simulation approach. Disagreement between measurement and Fuchs’s approach when applied to the ion-ion recombination has been known for decades.
In addition to the incorrect comparison, the statements about Tamadate et al are also largely inaccurate:
“ Thus, they restricted the MD simulation to the limiting sphere while using the continuum (diffusion) equations outside the limiting sphere….” They actually use a cubic simulation domain of gas molecules that follows both ions. Simulations do not necessarily use Fuchs’s definition of the limiting sphere, but they adjust the sphere radius used as the boundary between continuum and MD to ensure that this radius is large enough not to influence results.
“The MD simulations were run for different conditions: with and without the influence of electrostatic forces,” Tamadate et al (2020) do not run simulations excluding electrostatic forces (which are extremely important in this problem). They do appear to include and exclude the initial electrostatic velocity for the incoming ions in the limiting sphere theory, but this is very different from including or excluding forces.
“In order to derive the recombination coefficient, they used two different approaches: the theory by Fuchs (1963) and the one by Filippov (1993).” Filippov’s 1993 approach is a more general version of Fuchs 1963 (and earlier) derivation. They are not different approaches. Tamadate et al (2020) very clearly uses Filippov’s equations and states this unambiguously. Tamadate et al do retrace the steps of Fuchs and Filippov, but I believe they appropriately credit where these steps come from.
“In Fig. 5 (e), the limiting sphere theories Na59, HF86, and Ta20 are shown. Whilst Na59 and HF86 agree fairly well with each other, Ta20 yields α values which are one order of magnitude too low (2.7 · 10–6 cm3 s –1 at ground level) and is, therefore, not recommended.” To reiterate, the plots displayed are not an accurate test of Ta20, as the probability of Fuchs was used. This statement is hence very inconsistent with the earlier statement in this manuscript, “While the approach of Tamadate et al. (2020) is very promising, they correctly emphasise the need for hybrid continuum-MD simulations with N2 and O2, instead of He, in order to achieve results comparable to atmospheric conditions.” The authors here have not made the appropriate comparison.
I also believe the authors are mistaken in the computational power and expertise required to perform such MD calculations. Certainly MD approaches need to be developed further to make use easier. However, it is not unfeasible to use MD calculations to compute and tabulate the ion-ion recombination rate under a variety of conditions. I do not agree with the statement “ Simulation experiments at temperatures and pressures representative of the different layers of the lower atmosphere could provide a better insight into the variation of the ion-ion recombination coefficient α in the atmosphere. Eventually, parameterisations are needed for everyday us because MD simulations require advanced computing power and experience.” The MD simulation approach the authors are discussing is only ~1 year old, and notion that this cannot become a common approach to compare to data, or even to predict the recombination rate in the future seems short-sighted and overly dismissive.
2. Second, the comparison to Hoppel & Frick (1986) is odd. Hoppel & Frick specifically developed a theory to describe the ionisation of particles, and use the ion-ion recombination coefficient as an input to bracket results (their concern was ensuring that the rate of small particle-ion recombination agreed with the ion-ion recombination rate and noticed that in Fuchs’s theory this would not be the case, so they worked rather hard to develop an approach taking the essence of Fuchs’s theory but which would converge to the ion-ion recombination rate). Stated differently, they use the ion-ion recombination rate as an input to their theory, not an output. To quote Hoppel & Frick: “The value of the recombination coefficient for atmospheric ions is here taken to be that given by Nolan (1943) as 1.4 x 10-6 cm3 s-1. For any value of ionic mass, a corresponding value of the ion-ion trapping distance d can be determined.” If the authors choose to compare to Hoppel and Frick, then I believe they should make clear for each comparison what the reference recombination rate being used is and what the temperature and pressure is for it- did they use the same as Hoppel and Frick of 4 x 10^-6 cm^3 s^-1 at atmospheric pressure and room temperature?
3. Based on comments 1 and 2, I do not agree with the “intercomparison” approach- this treats various theories as fixed and isolated approaches from one another, as opposed to bodies of work building off one another. Rather than perform an intercomparison of different theories where inputs are selected in advance and the theory is determined to be applicable to the data or not, I believe a healthier approach would be to use the data presented to determine the most ambiguous parameters in theories. For example, in the case of limiting sphere theories, the most appropriate thing to do would be to determine p(δ) in equation (37), the probability needed to find agreement with experimental data. This would be much more useful than an intercomparison, and would enable the authors to discuss how this probability varies. Similarly, the authors can determine the value of “d” needed in equation (26) for agreement with data. I believe Tables of p(δ) and d for different temperatures, pressures, and relative humidities would be quite useful and referred to extensively by others. I strongly suggest the authors to adjust their approach to provide such tables, as opposed to an intercomparison approach which is skewed by subjective choices in inputs.
4. I would also encourage the authors to expand the data set they use in comparison. There is no reason to limit to atmospheric air when comparing theories.
5. The authors do neglect the recent equations of Chahl & Gopalakrishnan (doi: 10.1080/02786826.2019.1614522) who focused on small nanoparticle-ion collisions, but their equations could be extended to ion-ion recombination easily.
Editorial Comments:
1. The line colors in most plots are too similar to one another, and I have a tough time linking the lines in plots to the legend.
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RC2: 'Comment on acp-2021-795', Anonymous Referee #2, 20 Dec 2021
The paper by Zauner-Wieczorek, Curtius, and Kueten discusses the history of ion-ion neutralization measurements and theory in the atmosphere. The authors have done a good job of digging up many old references, some of which are new to me and my group although we work in this field. However, I and other members of my group have some serious issues with the paper. I wonder if all this detail on the chemical physics of the process is completely lost on atmospheric chemists.
The manuscript is very hard to read with lots of jargon, many references to various studies that are hard to keep in ones mind. The manuscript is full of confusion between total recombination rate constants without separating what refers to two body and three body contributions are added. I would start with the simpler story of two body recombination and start adding three body processes in the introduction. I realize that some of the early work measures the total rate but going back and forth is difficult so that one compares things that shouldn’t be compared.
The early history of neutralization is interesting and worth noting, but in my opinion not worth all the detail and equations given in the manuscript. While early researchers like Thomson knew that ion-molecule reactions apparently take place, actual rate constants weren’t measured until the 1950s and 1960s, meaning that early researchers couldn’t appreciate the complexity of the ion types that were actually involved in the neutralization. It’s my opinion that the quantitative similarity of early measurements is mostly a coincidence. Even Loeb in his later books said that it was only after WWII that electronics were advanced sufficiently to make decent measurements. (This from memory; it would take some time to find exactly what Loeb said.)
The manuscript is mainly concerned with 3-body neutralization, but it seems to me that binary and ternary measurements or theory are not well distinguished, for example, they are mixed in Table 2. An example I know something about: both the Hickman (incorrectly evaluated) and Miller expressions are plotted vs altitude in Fig. 4(a) even though both are solely for binary neutralization and completely inappropriate for a plot vs altitude.
The 3 body work of Smith and Adams has been questioned by Rainer Johnsen.
Beginning with line 255, the results of Hickman are quoted incorrectly. The formula in Eq. (24) is Hickman’s, however, Zauner-Wieczorek et al. says that Hickman’s reduced mass is in amu, but that’s not right; Hickman used reduced masses in atomic units (the mass of the electron). Use of the formula as stated by Zauner-Wieczorek et al. would lead to rate constants 200 times too large.
Further, some particular data are incorrectly quoted from Hickman’s paper. It’s important to note that those data were not Hickman’s. He was using data from the SRI merged beams experiments. It is now known that the SRI molecular ions were highly vibrationally excited (if not electronically excited), as was later shown by the SRI people themselves with a new collinear ion-laser experiment, which is the reason no further measurements were made with their merged beams apparatus. The important point is that the data quoted are incorrect because Zauner-Wieczorek et al. assumed that the units were E-06 cc/s, but Hickman clearly states that the units are given in Fig. 4, where E-08 cc/s is stated. The same units are specified in Fig. 3 along with the units for m (atomic units).
Beginning with line 260, the results of Miller are quoted incorrectly. The quoted formula is the same as Hickman’s except that the reduced mass is given in amu instead of Hickman’s atomic units. So the formula should not be attributed to Miller. Miller used flowing afterglow data that existed at that time (1979) to improve on Hickman’s parameterization instead of using the faulty SRI merged beams data. The formula developed by Miller is not quoted by Zauner-Wieczorek et al., namely, a = 3.32E-07 (T/300)-0.5 m-0.52 EA-0.24. The “T<1000K” is a limit imposed because the neutralization cross section is known to depend on 1/E at least for such temperatures.
The results of Hickman and Miller are consequently misstated in Table 2, and even worse in Fig. 3, where Hickman’s rate constants lie two orders of magnitude above Miller’s. Surely this discrepancy should have tipped off one of the authors to reexamine those two papers. Fig. 4 is likewise misleading.
The 1980 paper of Miller is only of historical interest and shouldn’t be considered in this manuscript at all. The type of analysis attempted by Miller in 1980 has been superseded by a more recent paper utilizing far more data: T. M. Miller, N. S. Shuman, and A. A. Viggiano, “Behavior of rate coefficients for ion-ion mutual neutralization, 300-550 K” J. Chem. Phys. 136, 204306 (2012), in which these parameterizations were given (m in amu, EA in eV):
a = (2.8 ± 1.0)E-07 cc/s (T/300)-0.9±0.1 m-0.5±0.1 EA-0.13±0.04 for polyatomic ions
a = (3.2 ± 1.4)E-08 cc/s (T/300)-1.1±0.2 m-0.01±0.09 EA-0.04±0.23 for diatomic ions.
Besides the more recent work of the AFRL group using the VENDAMS technique to derive the above parameterizations, they also miss exciting new work from the Urbain Group and DESIREE group in Stockholm. Also the Prague group has done the most fundamental work on three body increases to the overall rate constants. No mention of product formation is mentioned. Not always is the process a simple electron transfer.
The electron affinity of NO3- is 4 eV not 1eV. Why is H3O+(H2O)3 represented by mass 150? I know the answer is other positive ions exist but that should be clear.
Given that the work we know well is misrepresented, we, of course, worry that more work has also been misrepresented.
This paper needs at least a major rewrite and I don’t believe it belongs in this journal.
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RC3: 'Comment on acp-2021-795', Anonymous Referee #3, 26 Dec 2021
The manuscript authored by Zauner-Wieczorek et al. presents a good review of the historical theory development on ion-ion recombination under relevant conditions of the troposphere and lower stratosphere. The authors then made a simple sensitivity study on the limiting sphere theories and compared the different parameterisations of the theories to measurement data from a few laboratory and field as well as model results. The content of the work, especially the review part, is valuable. The comparison studies are a bit flimsy, without discussions on why some parametrisations worked poorly and there was no insights given for corrections or improvements. The clarity of the manuscript needs to be improved and the manuscript needs somewhat a major revision.
Comments:
When talk about ion-ion recombination, could you please first of all provide the definition of ion? Do you also consider the recombination of charged aerosol particles as ion-ion recombination?
You compared the different parameterisations on ion-ion recombination to a few laboratory, field and model results and demonstrated that some models clearly have poor performance but did not discuss the potential causes. Could you please elaborate on this and provide insights into how they may be corrected or further improved?
Based on the comparisons with laboratory, field and model data, you suggested Brasseur and Chatel 1983 over other parameterisations. Given the fact that it has the semi-empirical nature, it is expected to agree better with measurement data. The measurement data (whether it is Rosen&Hofmann, Gringel et al., Morita or Franchin et al.) are based on probing air ion concentrations. Air conductivity is intrinsically dependent on ion concentration. Then the uncertainty from measurement loss inside the instrumentation or the system cannot be avoided. This was not discussed in the manuscript when making suggestions on the choice of theory.
You did not recommend Tamadate 2020 due to its resulting in large deviation from measurement data. It seems however that the authors did not perform a MD simulation as described in Tamadate et al. 2020, instead the authors used the formula listed in Table 2 and referred that as Tamadate 2020. However, this functional form is merely Filippov's approach, which is similar to Fuchs model, as described in Tamadate 2020.
I also find the manuscript was not very carefully prepared. The notations are especially confusing. For example, the mathematical symbol of prime should be used instead of ’ (e.g. p6 L137). Also d have several definitions through the manuscript, which is confusing. v+ and v- were not defined where they appear first and definitions of U+ and U- in eq 8 were missing. It is also unclear what is x on p5 L128. A few different notations were used for the same property, e.g. e and eT for collision probability, d and d3 for three-body trapping distance, etc. It is also sometimes difficult to distinguish between similar symbols like a and a and M for molar mass and [M] for number density of air molecules. Please revise the manuscript carefully and drop off the repeated notions and use symbols that can be better distinguished.
p7 L160-161. It is confusing that you talk about 'collision probability becomes almost 0' and then 'collision is governed by the collision cross section'. Could you please elaborate what you mean here? How do you distinguish 'collision probability' and 'collision cross section'? To my understanding, the CCS is just a different way to quantify the probability of successful collisions.
p7 L177. normal value? what is not normal?
P13 L367. what do you mean by 'ion current'?
p18 L472. what do you mean by 'trapping sphere'? Is it different from limiting sphere?
p24 L587. Ta20 yields α values which are one order of magnitude too low (2.7 · 10–6 cm3 s–1 at ground level). Is it true? 2.7e-6 cm3s-1 does not seem too low.
Fig.1 caption. please consider using open circle instead of white point.
Fig.3c The color for Tamadate et al. 2020 in legend is different from that in the plot.
Table 1. please define the symbols in the caption. what is r0?
I also suggest that you consider restructuring some parts of the text. I find organisation of section 2 in the current manuscript does not render a smooth textflow, especially concerning the definition of d. Because d appears earlier in the text already but its definition comes quite late. Also in section 4, there is a sudden jump to ion-aerosol attachment without preparing the readers with the purpose.
Marcel Zauner-Wieczorek et al.
Marcel Zauner-Wieczorek et al.
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