Micro-Kinetic Modelling of CO-TPD from Fe(100)—Incorporating Lateral Interactions
Round 1
Reviewer 1 Report
This manuscript pertains to the microkinetic modelling of CO-TPD over a Fe(100) surface wherein the effect of directly incorporating lateral interactions into mean-field microkinetic modelling has been explored.
This is a very interesting study, clearly relevant to researchers working in the field of microkinetic modelling (both mean-field as well as kMC). The article is well-written, the results and reasoning are clear and the article is pleasant to read. Furthermore, the theoretical methods employed are state-of-the-art.
I have a few suggestions to further strengthen the article. I kindly invite the authors to consider these. I recommend this manuscript to be published in Catalysts after some minor changes.
1. In Table 3, it is not clear in the text or the caption where the values for the pre-exponents are derived from. The pre-exponents are (presumably) temperature-dependent, but this is not mentioned in the text.
2. What I find missing in this paper is a comparison between the mean-field models and kMC, and which of the models fits best with experiment. Could the authors perhaps expand their discussion / conclusion section and perhaps give a recommendation to the reader which method is preferred or a reasoning under which conditions one method is more advantageous over the other?
3. Line 49 states "... Fe(100). This system is of interest due to its relevance to the industrial Fischer-Tropsch process and may thus yield insight in the importance of lateral interactions under those conditions." The active phase of Fe-based FTS is commonly accepted to be some form of carbide. To what extent is the metallic Fe(100) surface of relevance to industrial FT conditions? Could the authors perhaps speculate in the discussion section how the results might change when subsurface or surface carbon is present?
4. The CO-CO lateral interactions are evaluated at a relatively low coverage of 0.25 ML. Could using this low coverage result in an underestimation of the lateral interactions? Can the authors give a prediction to how the results might change had the lateral interaction energies been determined at higher coverages?
5. Paragraph at line 210: it is not entirely made clear why the Bragg-Williams approximation is expected to perform better for systems with low mobility compared to the quasi-chemical approach working better at very high mobilities. Which of the two mean field models is expected to work best for TPD?
6. Line 232 states "here QCO(g) and QCO* are partition functions for CO in the gas phase and adsorbed state, respectively". What do these partition functions consist of and how were these partition functions determined? For example, does QCO* only contain vibrational contributions, or is there a rotational component included? Assuming that QCO(g) consists of translational, rotational, and vibrational contributions, how were these contributions determined? A possible method would be to use thermodynamic tables to get the QCO(g) as has been done by Zijlstra et al. (https://doi.org/10.1016/j.cattod.2019.03.002).
7. Coverage dependent activation barriers are obtained by a BEP correlation (line 262). The BEP factor alpha is claimed to be 0 for early transition states and 1 for late transition states. The authors should rephrase this to indicate that these are limits, rather just two options. Can the authors provide proof (or an indication) that the BEP correlation is valid for systems with varying lateral interactions? The authors should at least mention that the BEP correlation is in principle based on the assumption that the curvature of the potentials does not change significantly between the systems. (https://pubs.acs.org/doi/10.1021/cr9001808)
8. In Figure 5 there seems to be some sort of circle forming in the coverage plots. What is this phenomenon? There appear to be two such circles at 360 K (left middle and lower right corner) and one at 600 K (lower right corner).
9. Can the authors mention in the section "Simulation setup" the type of ODE-solver used and the absolute and relative tolerances? If the ODE integration method uses a Jacobian, how was this Jacobian constructed?
Spelling errors and other minor issues:
168: Table 2, orient the column "configuration" to be not vertical, but horizontal like the rest of the table.
210: The remainder of the sentence seems to have been lost, it ends suddenly.
223-225: In the diffusion steps, the adsorbent place does not move in the equation. It may be more clear to have CO* + * -> * + CO*.
233: Ed(theta) does not correspond with the form in the formula in 231.
253: Caption of Table 3: "As reported by [6] a unit?" Sentence seems incomplete.
308: show -> shows
543: Inconsistent naming. Fe(001) -> Fe(100)
545: For kMC -> For the kMC
553: necessary -> necessarily
556: Remove the sentence: "This section is not mandatory, but can be added to the manuscript if the discussion is 556 unusually long or complex."
558: Consistent abbreviations: EvS (line 558) and EVS (line 560)
Author Response
We thank Reviewer 1 for his/her constructive comments on our manuscript and have taken his comments into account in our revised manuscript. In detail
1. In Table 3, it is not clear in the text or the caption where the values for the pre-exponents are derived from. The pre-exponents are (presumably) temperature-dependent, but this is not mentioned in the text.
From the rate expressions we gave for surface reactions and adsorption-desorption reactions, the terms excluding exp(-Ea/RT) were considered to be the pre-exponential factors. In all cases, both BW/QCA and kMC, the pre-exponential factors were treated as temperature dependent inputs.
We have clarified this in the text, by adding
Hence, the pre-exponential factors in all our models are dependent on temperature. It should be noted that in principle lateral interactions also affect the pre-exponential factor due to a change in the vibrational partition function seeing its relative small contribution in the change in the partition function.
2. What I find missing in this paper is a comparison between the mean-field models and kMC, and which of the models fits best with experiment. Could the authors perhaps expand their discussion / conclusion section and perhaps give a recommendation to the reader which method is preferred or a reasoning under which conditions one method is more advantageous over the other?
We have added the following to our discussion section:
The experimentally observed CO-TPD profile from Fe(100) [18] seems to be modelled best using either the quasi-chemical approach (QCA) or the kinetic Monte Carlo method (kMC). The quasi-chemical approach is a mean-field model assuming a rapid equilibration between the interacting species. The kinetic Monte Carlo method accounts for both local effects and finite rate of diffusion. Reducing the activation energy for the diffusion of CO did not change the low temperature region of the simulated CO-TPD profile using kMC significantly implying that the origin of the low temperature desorption maximum in the kMC model is due the local effect. (i.e. the presence of CO still surrounded by co-adsorbed CO at the nearest neighbour position, which have a relatively low activation energy for desorption). The local effect simulated in the kMC model is further demonstrated by slowing down the diffusion of in particular surface oxygen (vide supra). The similarity between the CO-TPD profile from Fe(100) simulated using the quasi-chemical approach (QCA) and the kinetic Monte Carlo method (kMC) may seem therefore fortuitous. However, the energy as modelled using the quasi-chemical approach (QCA – see eq. 8) is in principle a sigmoid function rather than a linear function as predicted by the Bragg-Williams approximation. Hence, the increase in the adsorption energy with decreasing coverage will be less in the QCA model compared to the BWA-model. Hence, QCA starts mimicking the behavior observed with kMC for systems with repulsive lateral interactions also for systems with low rate of surface diffusion. Hence, QCA appears to be the next best approach if the required parameter set for kMC is not available or if the kMC-model becomes computationally too cumbersome.
3. Line 49 states "... Fe(100). This system is of interest due to its relevance to the industrial Fischer-Tropsch process and may thus yield insight in the importance of lateral interactions under those conditions." The active phase of Fe-based FTS is commonly accepted to be some form of carbide. To what extent is the metallic Fe(100) surface of relevance to industrial FT conditions? Could the authors perhaps speculate in the discussion section how the results might change when subsurface or surface carbon is present?
The reviewer here is correct that iron-based Fischer-Tropsch catalysts operate as iron carbides in the low temperature region (there is still some debate around the high temperature Fischer-Tropsch synthesis under industrial conditions). Nevertheless, metallic iron has been used over the years as a model system to gain some insights into the Fischer-Tropsch process. Here, we use CO-TPD from Fe(100) primarily as a tool to ‘calibrate; micro-kinetic modelling based on known and experimentally measured CO-TPD profiles. We do not think that extending the discussion on possible CO-TPD from iron carbide would be of interest.
4. The CO-CO lateral interactions are evaluated at a relatively low coverage of 0.25 ML. Could using this low coverage result in an underestimation of the lateral interactions? Can the authors give a prediction to how the results might change had the lateral interaction energies been determined at higher coverages?
The lateral interaction of CO-CO were studied in detail at a coverage of 0.25 ML. However, these interaction energies were compared with DFT-calculated adsorption energies at much higher coverage, viz, 3/16, 0.25 and 0.50 ML (see Fig. 1b) We show that the predictive power of the adsorption energy due to differences in the CO-CO lateral interactions is good. In order to clarify this to the reader, we altered the figure text for figure 1:
Figure 1. CO adsorption on p(4x4)-Fe(100): (a) ΘCO=0.25 ML in different configurations and their adsorption energy (details given in Table 1); (b) Testing the empirical relationship (see text) to determine the adsorption energy in various configurations at ΘCO = 3/16 ML, 0.25 ML and 0.5 ML (black: empirically determined adsorption energy; red: DFT determined adsorption energy).
It should be further noted that we performed an extensive, sensitivity analysis on the CO-CO lateral interaction to see how it affects the CO-TPD profile
5. Paragraph at line 210: it is not entirely made clear why the Bragg-Williams approximation is expected to perform better for systems with low mobility compared to the quasi-chemical approach working better at very high mobilities. Which of the two mean field models is expected to work best for TPD?
In the absence of surface diffusion, the probability for finding a species at the next neighbour site is given by the fractional coverage, since the species cannot move towards energetically more favorable sites. The manuscript has been amended to reflect this:
This kind of model may work well for systems with a low or negligible mobility of the adsorbed species (either through diffusion or via desorption and re-adsorption) resulting in a ‘frozen’ surface structure, since surface species cannot diffuse to an energetically more favorable state, although the .
6. Line 232 states "here QCO(g) and QCO* are partition functions for CO in the gas phase and adsorbed state, respectively". What do these partition functions consist of and how were these partition functions determined? For example, does QCO* only contain vibrational contributions, or is there a rotational component included? Assuming that QCO(g) consists of translational, rotational, and vibrational contributions, how were these contributions determined? A possible method would be to use thermodynamic tables to get the QCO(g) as has been done by Zijlstra et al. (https://doi.org/10.1016/j.cattod.2019.03.002).
The partition function for surface species only has a vibrational partition function, whereas CO in the gas phase also has contributions from the translational and rotational partition functions. The following sentence has been added:
(here QCO(g) and QCO* are partition functions for CO in the gas phase and adsorbed state, respectively, taking into account the vibrational and for CO in the gas phase also the translational and rotational partition function
with and are partition functions for the transition state and initial state, respectively, only considering the vibrational contribution
7. Coverage dependent activation barriers are obtained by a BEP correlation (line 262). The BEP factor alpha is claimed to be 0 for early transition states and 1 for late transition states. The authors should rephrase this to indicate that these are limits, rather just two options.
We agree and have amended the manuscript accordingly:
The factor α is in the limit case 0 for reactions with an early transition state, such as CO dissociation, and 1 for reactions with a late transition state, such as C/O recombination
Can the authors provide proof (or an indication) that the BEP correlation is valid for systems with varying lateral interactions? The authors should at least mention that the BEP correlation is in principle based on the assumption that the curvature of the potentials does not change significantly between the systems. (https://pubs.acs.org/doi/10.1021/cr9001808)
This is an interesting and important suggestion, which should be probed going forward as a fundamental question regarding Hammond’s postulate. We have amended the manuscript as follow:
The coverage dependent activation barriers can be obtained by invoking the Brønsted−Evans−Polanyi correlation and Hammond’s postulate [34], i.e. implying that the curvature of the potential energy surface does not change significantly due to lateral interactions [35]:
8. In Figure 5 there seems to be some sort of circle forming in the coverage plots. What is this phenomenon? There appear to be two such circles at 360 K (left middle and lower right corner) and one at 600 K (lower right corner).
This is a natural consequence of lateral interactions: repulsive lateral interactions will result in a lower activation energy for desorption and more stable species (species with a higher activation energy for desorption) will remain on the surface. This may result in structures (curves) on the surface, which have minimal or even attractive lateral interactions
9. Can the authors mention in the section "Simulation setup" the type of ODE-solver used and the absolute and relative tolerances? If the ODE integration method uses a Jacobian, how was this Jacobian constructed?
The following has been added to the methodology section:
All spelling errors and other minor issues have been addressed
Author Response File: 
Author Response.pdf
Reviewer 2 Report
The article entitled “Micro-kinetic modelling of CO-TPD from Fe (100)-incorporating lateral interactions” submitted by Gambu et al. present an interesting theoretical study of the CO interaction with Fe surfaces analysing the effect of the lateral interactions. Although the adsorption of CO on Fe(100) surfaces has been previously studied, I consider that this work provides additional insights on and it is susceptible for publication in Catalysts. However, prior to acceptation some minor corrections must be considered.
1) INTRODUCTION: In my view, the paragraph of lines 48-51 should appear after that paragraph of lines 52-61.
2) If “ML” corresponds to a volume unit, it should be rewritten correctly in the form “mL” along the text
3) In line 209-210, the phrase is incomplete: ”….resulting in a frozen surface structure, although the …???”
4) Line 219: CO-dissociation reaction contains a mistake (CO* against O*)
5) The conclusions are too concise. It could provide more information. Furthermore, the sentence “This section is not mandatory, but can be added to the manuscript if the discussion is unusually long or complex” likely coming from the template must be omitted.
6) The theoretical results are very interesting. Have performed the authors any experimental study to corroborate these data?
Author Response
We thank Reviewer 2 for his constructive comments and have considered his suggestions as follows:
1) INTRODUCTION: In my view, the paragraph of lines 48-51 should appear after that paragraph of lines 52-61.
We agree and the paragraph has shifted in the revised manuscript.
done
2) If “ML” corresponds to a volume unit, it should be rewritten correctly in the form “mL” along the text
ML corresponds to Monolayer coverage; has been formally defined at the first position it appears
‘of CO on p(4x4)-Fe(100). At a coverage of CO of 0.0625 monolayer (ML),’
3) In line 209-210, the phrase is incomplete: ”….resulting in a frozen surface structure, although the …???”
We rephrased this sentence:
This kind of model may work well for systems with a low or negligible mobility of the adsorbed species (either through diffusion or via desorption and re-adsorption) resulting in a ‘frozen’ surface structure, since surface species cannot diffuse to an energetically more favorable state, although the .
4) Line 219: CO-dissociation reaction contains a mistake (CO* against O*)
Noted and changed:
CO-dissociation (2)
5) The conclusions are too concise. It could provide more information. Furthermore, the sentence “This section is not mandatory, but can be added to the manuscript if the discussion is unusually long or complex” likely coming from the template must be omitted.
We have expanded the conclusion section as follows (also in conjunction with comments made by Reviewer 1):
The resulting models using either quasi-chemical approach (QCA) or kinetic Monte Carlo (kMC) result in the appearance of a separate low temperature desorption peak. The desorption peak at low temperature in the kinetic Monte Carlo method is caused by local effects, i.e. the presence of significant amounts of CO present of Fe(100) with multiple CO on next nearest neighbouring sites, since surface diffusion does not appear to affect the appearance of this desorption peak. The appearance of the low temperature desorption peak in the simulated CO-TPD profile from Fe(100) is related to the sigmoid function describing the adsorption energy as a function of coverage. This section is not mandatory, but can be added to the manuscript if the discussion is unusually long or complex.
6) The theoretical results are very interesting. Have performed the authors any experimental study to corroborate these data?
The study provides an explanation for the observed, experimental data reported by Moon et al. (1985).
Author Response File: Author Response.pdf
Reviewer 3 Report
The paper nicely illustrates several approaches to model TPD spectra from DFT. The main weakness is the simplicity of the adsorption energy model. I also miss a comparison with experimental TPD spectra. Overall, this is an instructive paper that warrants publication.
Detailed comments:
1. The relevance of Fe(100) for Fischer Tropsch is limited since the actual catalyst is iron carbide.
2. In the energy equations, the interaction parameters are estimated from only 4 configurations and the contributions are assumed to be additive (the repulsion of a CO interacting with 4 CO’s or C’s in neighboring sites is assumed to be 4 times that repulsion), and interactions with C and O are also assumed to be additive.
The reason for the attractive NNN interaction is also not explained. Also, in my experience, such (nice but simple) estimates are rarely exact across all configurations, and some errors should be expected that the authors can easily provide via a handful of extra DFT calculations for a preferably random selection of extra configurations. In particular, if you based your “empirical observation” on a few different sets of unique configurations, how much would your energetic parameters change? Also, what exactly is the estimated error in predicted adsorption energy for the configurations that were already used (and any extra you calculate)? These errors may be comparable to the energetic parameters themselves.
3. Section 2.2. A detailed O/Fe(100) model was reported by Bray et al. (Top. Catal. (2018)) and the author’s results could be compared with this.
4. No effort is made to compare the TPD spectra with experimental spectra. Are there any conclusions concerning which method provides the best match to experiment? It would also be nice to see more discussion on how well each mean-field model holds up against the more accurate kMC: perhaps the authors could discuss which they think would be best to use moving forward if one were unable to use kMC.
5. Line 210. Part of the sentence is missing.
6. Figure 2. Why are the units in “a.u.” when conversion to ML/s can be done?
7. Does the 4x4 unit cell provide enough flexibility to sample the possible stable ordered configurations? Are ordered configurations known for CO on Fe(100), e.g., from LEED?
8. How was the deformation energy (mentioned at several places) defined? The definition at the bottom of Table 1 is not clear – why is carbon included in the formula for CO adsorption?
9. Eq. (2) is not balanced, it has to be CO* + *
10. Eq. (4): the entropy gain during desorption could increase the rate coefficient.
11. Line 374: The statement that CO does not adsorb at bridge site is not tested.
Author Response
We thank the reviewer for his/her detailed comments
1. The relevance of Fe(100) for Fischer Tropsch is limited since the actual catalyst is iron carbide.
We agree with the reviewer that conclusions obtained from studied on Fe(100) cannot be used directly to evaluate the performance of iron carbide in the Fischer-Tropsch. Nevertheless, work on this system has been performed over the last decades to gain some insights on the interactions of this model system. The primary aim of this study was to see to what extent lateral interactions may explain the experimental CO-TPD from Fe(100), which was doeractio
2. In the energy equations, the interaction parameters are estimated from only 4 configurations and the contributions are assumed to be additive (the repulsion of a CO interacting with 4 CO’s or C’s in neighboring sites is assumed to be 4 times that repulsion), and interactions with C and O are also assumed to be additive.
As indicated in Figure 1, we tested the simple CO-CO repulsion on a set of different configurations and obtained a reasonable predictive power for CO-CO-repulsion
The reason for the attractive NNN interaction is also not explained.
A part of the attractive NNN interactions originate from a change in the deformation energy, but there is some contribution from through surface interactions as well.
Also, in my experience, such (nice but simple) estimates are rarely exact across all configurations, and some errors should be expected that the authors can easily provide via a handful of extra DFT calculations for a preferably random selection of extra configurations. In particular, if you based your “empirical observation” on a few different sets of unique configurations, how much would your energetic parameters change? Also, what exactly is the estimated error in predicted adsorption energy for the configurations that were already used (and any extra you calculate)? These errors may be comparable to the energetic parameters themselves.
We agree with the reviewer that our interaction parameters only give a first insight into lateral interactions in this particular system, which can be expanded to 1-2-3 and 1-2-3-4 interactions. However, even this simple system may have a multitude of different interactions, which may need to be taken into account. We therefore used a simplified form with a single interaction energy and performed a sensitivity analysis on the extent of lateral interactions.
3. Section 2.2. A detailed O/Fe(100) model was reported by Bray et al. (Top. Catal. (2018)) and the author’s results could be compared with this.
We have taken note of the modelling study reported recently by Bray et al. and their incorporation of lateral interaction work will be considered further,
4. No effort is made to compare the TPD spectra with experimental spectra. Are there any conclusions concerning which method provides the best match to experiment? It would also be nice to see more discussion on how well each mean-field model holds up against the more accurate kMC: perhaps the authors could discuss which they think would be best to use moving forward if one were unable to use kMC.
The following section has been added to the manuscript:
The experimentally observed CO-TPD profile from Fe(100) [18] seems to be modelled best using either the quasi-chemical approach (QCA) or the kinetic Monte Carlo method (kMC). The quasi-chemical approach is a mean-field model assuming a rapid equilibration between the interacting species. The kinetic Monte Carlo method accounts for both local effects and finite rate of diffusion. Reducing the activation energy for the diffusion of CO did not change the low temperature region of the simulated CO-TPD profile using kMC significantly implying that the origin of the low temperature desorption maximum in the kMC model is due the local effect. (i.e. the presence of CO still surrounded by co-adsorbed CO at the nearest neighbour position, which have a relatively low activation energy for desorption). The local effect simulated in the kMC model is further demonstrated by slowing down the diffusion of in particular surface oxygen (vide supra). The similarity between the CO-TPD profile from Fe(100) simulated using the quasi-chemical approach (QCA) and the kinetic Monte Carlo method (kMC) may seem therefore fortuitous. However, the energy as modelled using the quasi-chemical approach (QCA – see eq. 8) is in principle a sigmoid function rather than a linear function as predicted by the Bragg-Williams approximation. Hence, the increase in the adsorption energy with decreasing coverage will be less in the QCA model compared to the BWA-model. Hence, QCA starts mimicking the behavior observed with kMC for systems with repulsive lateral interactions also for systems with low rate of surface diffusion. Hence, QCA appears to be the next best approach if the required parameter set for kMC is not available or if the kMC-model becomes computationally too cumbersome.
5. Line 210. Part of the sentence is missing.
Sentence has been completed:
This kind of model may work well for systems with a low or negligible mobility of the adsorbed species (either through diffusion or via desorption and re-adsorption) resulting in a ‘frozen’ surface structure, since surface species cannot diffuse to an energetically more favorable state, although the
6. Figure 2. Why are the units in “a.u.” when conversion to ML/s can be done?
The arbitrariness of the units are depicted here and the profile can be easily interpreted by the reader. The reviewer is correct that this can be easily displayed as well as ML/s (which may confuse some readers – see comments reviewer 2)
7. Does the 4x4 unit cell provide enough flexibility to sample the possible stable ordered configurations? Are ordered configurations known for CO on Fe(100), e.g., from LEED?
The ordered c(2x2) structure of CO on Fe(100) is known, which can be easily obtained on Fe(100). However, the objective of the study was to obtain an estimate for lateral interactions rather than probe all possible configurations of CO on Fe(100)Larger superstructures. Hence, the study was limited to a 4x4 cell.
8. How was the deformation energy (mentioned at several places) defined? The definition at the bottom of Table 1 is not clear – why is carbon included in the formula for CO adsorption?
The following has been added to the methodology section:
The strength of adsorption of CO was investigated in different configurations on a p(4x4) cell and was determined from the difference in the electronic energy of the system with adsorbed CO and CO in the gas phase and a bare surface. The value was normalized with respect to the number of CO molecules adsorbed per unit cell
(10)
Adsorption also results in a change in the position of the metal atoms in the slab. The deformation energy was determined by removing the adsorbates from the optimized geometry and determine the electronic energy of this systems relative to the electronic energy of the bare Fe(100) surface:
(11)
The deformation energy is dependent on the cell size. Lateral interaction can be described in terms of an excess energy, which was evaluated relative to individually adsorbed species at Θ = 0.25 ML: (12)
9. Eq. (2) is not balanced, it has to be CO* + *
The reviewer is correct and Equation (2) has been changed:
CO-dissociation (2)
10. Eq. (4): the entropy gain during desorption could increase the rate coefficient.
We formulated the rate coefficient for desorption in a thermodynamically consistent manner over the rate constant for adsorption and the equilibrium constant for adsorption. Hence entropy gain during desorption has been accounted for.
11. Line 374: The statement that CO does not adsorb at bridge site is not tested.
We (and others, see e.g. Sorescu et al., 2002; Bromfield et al., 2005) have shown the bridge sites form a local minimum (see the table below from P. van Helden, PHD thesis, University of Cape Town 2008) or is even unstable.
Data P. van Helden, PhD thesis, University of Cape Town, 2008.
Author Response File: Author Response.pdf
