Corresponding States for Volumes of Elemental Solids at Their Pressures of Polymorphic Transformations
Round 1
Reviewer 1 Report
In the paper it is shown that all known non-molecular elemental solids follow a rule of corresponding states if only the volumes and pressures of the polymorphic transitions are considered. This observation extends to high-pressure atomic polymorphs of elements that are molecular at ambient conditions. In my opinion, the content of the paper is significant enough for publication in Crystals; however, I believe that there are very significant issues that must be addressed before the work is considered acceptable.
The author claims that only experimental data are used, and the relation presented is fully empirical, without any kind of simplifying assumptions or theory. This is a limit of the work. I would like to have a theory for this behavior. Moreover, the author should increase the comments about the relevance of this work.
My other concern is the fact that Crystals is a journal with a very wide audience. Therefore, the introduction and the discussion of the results should be “for all” and not only for experts in polymorphic transformations.
In the caption of Figure 1, the author writes that “the existence of five distinct sets is clearly visible”; this is not true. I would like something in the plot evidencing this existence.
I strongly suggest the author extend the introduction and the result discussion taking into account my criticisms, with a clear aim of articulating what they want the reader to know most from this article.
Minor: the author should control the formula in the abstract and the presence of numbers in the keywords.
Author Response
In the paper it is shown that all known non-molecular elemental solids follow a rule of corresponding states if only the volumes and pressures of the polymorphic transitions are considered. This observation extends to high-pressure atomic polymorphs of elements that are molecular at ambient conditions. In my opinion, the content of the paper is significant enough for publication in Crystals; however, I believe that there are very significant issues that must be addressed before the work is considered acceptable.
Response (indicated by button marks):
- Thank you for your comments
The author claims that only experimental data are used, and the relation presented is fully empirical, without any kind of simplifying assumptions or theory. This is a limit of the work. I would like to have a theory for this behavior.
- I’m glad that you raise this point! Indeed, implementation of a theoretical background was considered but thought to be too long. Following your recommendation following sections were added (please note that the formulas cannot be reproduced here but are available in proper formating in the revised manuscript):
4.1 Theoretical explanation
‘Upward’ and ‘downward’ transitions between different states Ni match changes in valence electron configuration, wherever such information has been obtained experimentally [10,15,17] or through calculation [14].
The fact that the volumes Ni are nearly equal to integer multiples of rB3 has a straightforward theoretical basis: It implies that the volumes that occur at polymorphic transitions of non-molecular elements are controlled by the number of valence electrons rather than by their interactions. Electron exchange- and Coulomb-interaction account for part of the deviations from equation (1) and from integer Ni-values, thus, they are of second order with respect to the volumes. In other words: equation (1) is the equivalent to an ideal equation of state for elemental solids, only that it does not address continuous pressure-volume changes but discrete changes that occur at the phase transitions.
This finding can be further quantified by considering the effect of the different valence orbital states: Based on the experimentally determined pressure-induced hybridization of d-states of the alkaline metals Rb, Cs, and K (with no observable d-state occupancy to 40 GPa) [10]. Figure 1 may be re-parametrized in terms of pressure-induced changes in the d-state occupancy at valence level. In particular, d-state occupancy of the elemental polymorphs can be correlated to the five sets N2 to N6 shown in Fig. 1 through their 0 GPa intercepts Ni. This is shown in Figure 3a. For instance the d-state occupancy of Cs increases from phase Cs-II to -VI from ~0.0 to ~0.6 and from ~0.2 to 0.4 electrons between Rb-II to –VI [10]. The high-P polymorphs of Li and Na have little or no occupied d-states in this diagram and according to earlier computational work [2,4,14]. The correlation between Dn(d) and Ni is linear, at least for polymorphs of K, Rb, and Cs (black squares in Fig. 3a): K-II, Rb-II,-IV, Cs-II, and Cs-III. Within uncertainty the slopes are equal. In terms of multiples of the electron elemental charge this linear correlation between d-state occupancy, Vtr, and L is:
Ni = Ni0 – 6.11(14)×10-11 × n(d) × e (2)
where Ni0 = Ni at n(d) = 0, the constant factor is given in m3/(C×at) , and e is the electron elemental charge. Equations (1) and (2) can be combined to
+
(3a)
where Nm is the value assumed at n(d) = 0, P is in GPa, V in m3 /at, and e is the electron elemental charge. The effect of d®p transfer on shifts into sets with higher Ni has been discussed above. Overall there are less data about pressure-induced changes in valence p-state occupancy (summarized in the supplementary Figure S2) but within uncertainties the relation between Vtr/ L and n(p) has the same slope as in (3a) though with opposed sign (consistent with the transfer into higher sets Ni whereas changes in d-state occupancy generally result in changes to lower Ni). Implementing n(p) into (3a), dividing through rB3 and using the approximation Ni rB3 ~ 10×m gives:
+ 66(2)
= +
(3b)
where pressure is given in GPa , the volumes Ni are taken as integer multiples 10× i Î IN (the set of natural numbers) of rB3 per atom, p £ n(d), the increments Dn(d) are taken as 0.151, hence: 66.1×0.151 = 10 d-electrons and accordingly 66(2). 0.09(1) = 6(1) for the p-electrons. The actual values Ni may be substituted for 10m. An additional term for f-electrons with factor -14 may be added.
Moreover, the author should increase the comments about the relevance of this work.
- Thank you! I see the point - We added to the introduction and the conclusion the following sections:
Introduction: In difference to gases whose states can be described by the ideal gas- or the van der Waals equations of state, solids seem to defy the concept of a general equation of state. The range of structures and properties as well as their changes upon compression appears too large to be comprised in a single formula.
Conclusion: These states, as defined in equation (1) are close to integer multiples of the cube of the Bohr radius. Therefore, volumes and pressures of polymorphic transitions of these solids are controlled by the number of valence electrons as specified in equation (3b). This finding is in accord with the commonly used effective single electron-based computational approaches for modeling elemental solids within the range of 0 to 0.5 TPa. However, equation (1) can be used for constraining elemental metal volumes for first principal and empirical potential based calculations, thus removing the weakest point in these approaches, the proper assessment of volumes.
- We also added the abstract. The statement: ‘Volume, and pressure of phase transformations of non-molecular elemental solids obey a universal relation of discrete corresponding states’ (in Conclusions) marks a pretty high level of relevance within the field!
My other concern is the fact that Crystals is a journal with a very wide audience. Therefore, the introduction and the discussion of the results should be “for all” and not only for experts in polymorphic transformations.
- We added to the introduction and the abstract (see above). Note that this paper is part of a Special Issue about phase transformations at high pressure! The paper addresses precisely the topic of this SI. When reading the articles that have already been published in this SI I do not see that the present manuscript lacks generality – rather, it addresses a broader audience than most of the already published papers in this SI.
In the caption of Figure 1, the author writes that “the existence of five distinct sets is clearly visible”; this is not true. I would like something in the plot evidencing this existence.
- This is probably a misunderstanding. We show the correlation fit as additional supplemental figure 1 in order to clarify this issue – the adjusted R^2 of the correlation is 99.85%! What more can you ask for?
In Fig 1a five separate linear trends of transition-volume data are clearly visible: Note that empirical data exhibit scatter as result from experimental uncertainties and from material specific deviations. However, you may consider that the shown correlation is about as good as that of the van der Waals equation as applied to some 100 compounds - and it is markedly better than the correlation between the Dulong-Petit limit and actual high-T heat capacities of solids. Yet, you would certainly not claim that the Dulong-Petit limit is not true or lacks evidence.
I strongly suggest the author extend the introduction and the result discussion taking into account my criticisms, with a clear aim of articulating what they want the reader to know most from this article.
- Thank you again for these comments. I added to the introduction and Conclusion sections to clarify our point. A section about the theoretical background has been added. The Discussion section has been restructured with subsections for sake of clarity. One additional Figure and two supplemental figures were added.
- Please note that this article is for a Special Issue about phase transitions at high pressure. Therefore, the topic and discussion is centered on this topic. When reading the articles that have already been published in this SI I do not see that the present manuscript lacks generality – rather, it addresses a broader audience than most of the other papers in that SI (I made this point already above).
Minor: the author should control the formula in the abstract and the presence of numbers in the keywords.
- Thank you, we corrected the keywords. The formula is the abstract has been formatted (in the uploaded manuscript)
Reviewer 2 Report
This manuscript reported the corresponding states for volumes of elemental solids at their 2 pressures of polymorphic transformations. The topic of the manuscript is unclear, and the logic is not good. In addition, it exists many mistakes in the present version, such as the absence of Figure 2, non-standard Figures, text-left, and so on. Considering its overall quality and significance, this manuscript is suggested for submission to other journals.
Author Response
Thank you for the comments - the responses are indicated by buttons.
This manuscript reported the corresponding states for volumes of elemental solids at their 2 pressures of polymorphic transformations.
The topic of the manuscript is unclear
- The comment is appreciated. The topic of the manuscript is stated in the title and abstract. If you feel that this is unclear, please state specifically the points that need clarification.
, and the logic is not good.
- The Discussion section has been restructured and subsections have been added for better clarity. Additional paragraphs have been added to the introduction, discussion, and conclusion section to make the paper more consistent and show the relevance of the results. Note: This paper reports observations based on experimental data. Therefore the main part of the paper is focused on and guided by the description of these observations. If this does not read as brilliantly as a treatise by Condillac I apologize but the required level of detail and the sequence of different, seemingly unrelated aspects cannot be avoided here and should not be omitted.
In addition, it exists many mistakes in the present version, such as the absence of Figure 2
- We corrected the Figure numbers. Thank you for pointing this out!
, non-standard Figures, text-left
- Thank you for pointing this out - we fixed these issues.
, and so on. Considering its overall quality and significance
- Conclusions state ‘Volume, and pressure of phase transformations of non-molecular elemental solids obey a universal relation of discrete corresponding states’
The reviewer may not be so familiar with the topic of the Special Issue, but for this field of research the conclusions from this article are quite significant!
If you see particular problems with the conclusions and the data, please let us know.
In order to address concerns about quality and significance specific comments are needed: Do you see that the conclusions are not supported by the data, why, and at what point?
, this manuscript is suggested for submission to other journals.
- This manuscript was invited as article for a Special Issue and we leave this decision to the Editor and Guest editors of this journal.
Round 2
Reviewer 1 Report
All my issues have been addressed; therefore the work is considered acceptable.
Author Response
Response:
Thank you for the second review and comment and for the critical comments in the first review – they have helped a lot to improve the paper.
Reviewer 2 Report
The revised manuscript still exists some mistakes. For example, why does it have two Figure 2 in the present manuscript? And, in the text, it use Fig. 1 and doesn't Figure 1. The Figures' vertical and horizontal coordinates are still irregular.
Author Response
Response: Thank you for the second thorough, critical reading and the comments: The Figure coordinates have been corrected by placing them inside the text borders. The figure heights are accordingly adjusted by the template.
- All abbreviations ‘Fig.’ etc have been replaced by ‘Figure’. ‘Fig(ure)1a’ etc has been replaced and the figure numbers have been corrected. Figure numbering runs 1,2, 3a,3b,4. I prefer to keep the numbering of Figure 3a and b, if possible.
- I checked spelling and wording and made a few corrections (line 12, 72, 83, 87, 121,130, 159)
- I centered the formulas but will need support by the technical editors with the final formatting.
