Theoretical Predictions for the Equation of State of Metal Nickel at Extreme Conditions

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This manuscript by Wu and collaborators addresses the equation of state (EoS) of nickel to extremely high pressure and temperature conditions. The authors calculated the volume of nickel up to 500 GPa and 3000 K using the first-principles computational method for the internal energies of Ni as a function of volume combined with a statistical thermodynamics approach with the partition functions calculations, called DIA by the authors, for pressure-volume-temperature P-V-T data. The authors evaluated the present DIA method by comparing it with previous experimental data in diamond-anvil cells (DAC) and another computational method with a quasi-harmonic approximation (QHA) method. The authors claimed the present DIA agrees with selected DAC data and is better than previous QHA simulations. However, I am concerned about unclear details of DFT calculations, inadequate interpretations for the comparisons, and insufficient information on the raw data obtained in the present simulations. The details of my comments are given as follows.

Major issues:

1. The magnetic moments and spin states of atoms affect the thermoelastic behaviors of Ni at 1 bar (https://doi.org/10.1007/BF02644859). The ferromagnetism in fcc-Ni is confirmed experimentally to 70 GPa at 300 K and expected to be sustained up to 250 GPa (10.1063/1.2434184 ), which is also supported by DFT calculations (10.1016/j.physb.2011.10.047). However, it is unclear how the authors incorporated the magnetism or spin state effects on the present DFT and DIA simulations. The spin moments of Ni atoms should affect the calculated internal energies and the partition functions. The authors should clearly describe how to consider the effects in the DFT and DIA simulations and exhibit the pressure and temperature decencies on the magnetic moment and its contribution to the present P-V-T data. If not, the present P-V-T cannot be compared with the fcc-Ni data in experiments.

2. In Section 3.1, the authors seem to misunderstand previous experimental data and incorrectly interpret the differences between the present simulations and the previous data set. The authors compared the present results with experimental data and then stated the differences in experimental data from the present simulations are due to non-hydrostatic conditions in DAC experiments. However, the statements are based on misunderstandings by the authors of previous papers. The authors claim the hydrostatic conditions of Pigott et al. are better than those of Campbell et al. (L161). However, the NaCl softer medium used in Campbell et al. generally supplies better hydrostatic conditions than MgO. The authors claim the differences of Hirao et al. from the present simulations originally from the non-hydrostatic conditions in DAC experiments. However, the statement of the pressures under non-hydrostatic and hydrostatic conditions (L172) is not based on evidence. In other words, the non-hydrostatic pressure effects on Ni volume in the previous DAC experiments are not straightforward, as claimed by the authors. This is because the pressure was determined from Pt volume under conditions similar to the sample Ni. Therefore, the difference in pressures on Ni samples from hydrostatic conditions depends on the pressures on Ni and Pt in DAC. In addition, Hirao et al. evaluated deviatoric stress on the Ni sample and Pt pressure marker, which showed the deviatoric stress on both materials was similar and much smaller than the experimental uncertainties. Indeed, the data by other DAC experiments to 150 GPa with a He hydrostatic pressure medium by 10.1103/PhysRevB.77.094106.

Considering the information mentioned above, the present DIA results should be evaluated first by compared with the DAC measurements at 300 K by 10.1103/PhysRevB.77.094106 and 10.1063/5.0074340 . In Fig. 3(c), the DIA results exhibit non-systematical differences from experimental data above 100 GPa. Though the DIA simulations agree with the data at 40-100 GPa by Campbell et al. and Pigott et al. (Fig. 3ab), the simulations cannot reproduce the measurements below 40 GPa. The authors explained the inconsistency in the low-pressure data as “attributed to the accidental error in the experiments (L158)”. This statement should be invalid without any supporting materials.

As described above, the authors compare the experimental results incorrectly. A more reasonable comparison shows that the present simulations cannot reproduce experimental measurements, whereas some data points coincide with the simulations. I recommend the authors treat previous experimental data more carefully and evaluate the present simulations without fixed ideas. Without the consistency of the simulations with measurements at 300 K, the arguments on high-temperature results could not be valid because the uncertainties of high-pressure experiments tend to arise at higher temperatures. When the authors evaluate the present simulations at high temperatures, I recommend starting from the comparison with 1 bar and high-temperature volume data, which are determined with much higher accuracy than high-pressure data.

 

3. It is hard to find raw data of the present DIA simulations in the manuscript. In L143, the authors describe the obtained P-V data as a “red solid line” in Fig. 2. Also, the authors describe the calculated volume in Table 1, which does not match with very dense volume data shown with a line in Fig. 2. Actually, I am surprised to see a huge number of volume points in Table 1. It looks like tough work f to run DFT calculations and obtain the internal energies at lots of volume points. Could the authors give more detailed descriptions of the simulation volume steps? Relating to the raw data points, the EoS fitting results should be exhibited with the used data points to evaluate the reliability of the EoS. In addition, the EoS parameters should be given with their uncertainties.

 

Other comments:

L27: The description is an overstatement because the Ni content is believed to be limited to ~5% in the Earth's core (2014 Treatise on Geochemistry 3 559-577).

L31: The references are not enough. The measurements with a He pressure medium have been reported to be 156 GPa by (PRB 78, 104102  2008). If the authors include Pigott et al., please add Campbell et al. and Hirao et al., using the annealing method with solid pressure medium or He medium.

L151: It is hard to see the calculated results by the authors. Or were the simulations performed to give such dense line dots? If not, the authors first exhibit raw data from the current simulations.

L158: What do the authors mean by "accidental errors"? The authors should clearly describe the origin of the errors with reasonable evidence.

L198 and Fig. 4: The error calculations for the DIA data should be invalid. The volume and temperature errors are for the experiments. These errors should not be used for the error estimations for the present data because the experimental error and the present simulation errors are independent. If no statistical or systematic errors are confirmed for the simulated DIA data, the authors should not give any errors.

L234: As commented earlier, which data or values are from present DFT and DIA simulations is confusing. Please clarify, describe, and display the raw data in the text and the table. When taking the title of Table 1, the authors performed DFT and PF calculations at volume points corresponding to the P-T points shown in Table 1. The reader wants to know how the authors can fix the PVT during the DFT and PF calculations.  

L303, 319: The authors can give the Debye temperature because the DIA method includes the internal energy changes from 0 K to a higher temperature. Such a data set easily provides the Debye temperature of Ni, even if it is as low as 200-400 K. Please address the issue.  

L314: The Debye temperature should depend on the materials, which range in much broader temperatures. Moreover, the Deby temperature should increase with pressure.

L327: The authors should give the errors of the fitted parameters.

Fig. 7: Again, please do not use lines when the data are not so dense.

L358: This wording sounds logically funny. This is only the case where the DIA data shown here are assumed to be the correct values.

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The manuscript presents a scientifically rigorous and valuable contribution to the field of high-pressure materials physics. The use of the Direct Integral Approach (DIA) to predict the equation of state (EOS) of nickel under extreme conditions is both novel and timely. The authors provide a thorough comparison with experimental data and other theoretical models (notably QHA), and the development of a simplified analytical model with only two parameters is a notable practical advancement. However, the manuscript requires minor revisions before it is suitable for publication:

1. The abstract should highlight the objective, methods, results, and conclusion more distinctly. In addition to being concise, it should be clear.
2. The paper states that DIA results are better than QHA in many cases, but does not always quantify why or how that difference matters in practical applications. For example, what is the implication of a ~4% deviation at 3000 K?
3. The "universal model" deserves more focus—are there conditions or materials where it fails?
4. Some references in the main text are overly informal or lack consistency: e.g., "Ref. [1]" vs. “Pigott et al. [1]”. Pick one style and maintain it.
5. The Introduction could benefit from slight improvements, particularly in clarifying the current limitations of existing EOS models and more explicitly stating the novelty of the DIA in this context.

Comments on the Quality of English Language

Several grammatical issues and awkward phrases are present throughout the text, which at times hinder clarity. A full professional language editing is strongly recommended. 

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

Manuscript under consideration is “Theoretical predictions for the equation of state of metal nickel at extreme conditions”. The authors provide deep studying the equation of state of pure nickel at high temperatures and pressures. There are some comments.

1. The scope of journal “Metals” is processing, structure and properties or functions of all kinds of metals. Since it is hard to imagine the working conditions for nickel at 3000 K and 500 GPa, it would be useful to provide the suggestion, at least hypothetically, as to where such working conditions could appear.
2. The goal of work should be clearly stated in the Introduction.
3. Scientific novelty and practical significance of findings should be clearly emphasized in Abstract and Conclusions.

Author Response

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Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

Reviewer comments

Manuscript ID: metals-3589028

Title: Theoretical predictions for the equation of state of metal nickel at extreme conditions

Journal: Metals

This work uses the Direct Integral Approach (DIA) to predict the equation of state (EOS) of pure nickel up to extreme conditions (3000 K, 500 GPa). The predictions are validated against existing hydrostatic experimental data and a universal two-parameter EOS model is proposed.

It is stated that the paper falls within the scope of the journal, but some points need to be corrected:

1- Keywords in a scientific paper should be concise, avoiding compound words, to ensure clarity and ease of searchability.

2- The authors cite their own previous works up to 8 times in a relatively short paper. The authors should reduce self-citations where not strictly necessary, and include more citations to independent, recent, and diverse studies.

3- Lack of Clear Statement of Novelty. What new development, discovery, or improvement has been achieved? How their approach differs from existing studies- including their own previously published work. Include more recent and independent references (published in the last 5 years) to frame the need for this study.

4- In the introduction: Limitations of PCM and CMF models are discussed without acknowledging their valid application domains.

5- The "universal model" is only tested on Ni. Broader validation across multiple material classes is missing.

6- The introduction neglects modern high-temperature EOS approaches (e.g., machine learning potentials, …).

7- The manuscript has frequent awkward phrasing and typos:

• “theoretical predicting” should be "theoretical prediction"
• “predications” should be "predictions"

The manuscript has many grammatical mistakes, awkward phrasing, and poor logical flow between sections.

8- Line 259: “…included in the section A show…” no section A in the manuscript.

9- In section 3.3, line 347: Statements such as "pressure depends mainly upon the volume" are qualitative. Authors should quantify the relative sensitivity of pressure to volume vs. temperature.

10- In the title (caption) of Figure 8, add a description for Figure 8(c) to clearly explain what it represents.

11- Lines 353-359: The discussion mentions pressure differences of "5 GPa" to "35 GPa" between their model and the MGD model but does not normalize this difference. A percentage error analysis would be more informative, especially at very high pressures.

12- Grammatical errors (e.g., "plan" instead of "plane", "expends" instead of "expands").

13- The conclusion as currently written is superficial, lacks critical evaluation, and overstates the contributions. It should be significantly rewritten to summarize key findings quantitatively, critically acknowledge limitations, avoid unsupported generalizations, and propose meaningful directions for future work.

In conclusion, while the work addresses an interesting and relevant topic, it suffers from significant shortcomings in justification and clarification. Therefore, I recommend that the manuscript undergo major revision before it can be considered for publication.

Author Response

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Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

I have carefully read the responses to my comments and the revised manuscript. I appreciate the additional work and materials by the authors. I was satisfied with most of the responses. However, some important points and data in Responses have not been incorporated into the revised manuscript. The Figure in Response 1 apparently exhibits that the spin and magnetic moments affect the present P-V-T data. Because fcc-Ni sustains the ferromagnetism at least to 70 GPa at 300 K (10.1007/BF02644859; 2007 App. Phys. Lett. 90, 042505;  2012 Physica B: Cond. Matt. 407, 330-334), the authors should not ignore the effects on simulated P-V-T data for comparing these with the experimental data. The Figure in Response 6 is also essential to represent the validity of the present results, especially at high temperatures, because the experimental data at 1 bar are more reliable than those at high pressures. Those data are essential to demonstrate the validity of the present AIMD and DIA simulations. Without the simulation of the P-V-T data considering the spin moments and the comparison at 1 bar, the manuscript can hardly represent the validity of the present conclusions.

Other points:
L337: I cannot agree with the statement on the Debye temperatures for “most of solids”. The Debye temperature is related to the phonon frequency of materials, which should be variable among solid materials, e.g., the Debye frequencies of Si and diamond which can be general solid materials but have relatively high phonon frequency, are ~650 K and ~2300 K. Also you can easily find Debye temperatures much higher than 400 K when taking a look at hard materials. I guess the authors mean “simple metals (or pure elements)” instead of “solids”.

Author Response

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Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

The authors have adequately addressed the comments raised in the reviewer report. Therefore, I recommend the paper for publication in the journal.

Author Response

Thanks for the review.