First-Principles Study of the Effect of Sn Content on the Structural, Elastic, and Electronic Properties of Cu–Sn Alloys

Round 1

Reviewer 1 Report

Introduction:

"However, currently, the calculation of copper tin alloys has been focused on the 55 specific phase structure of copper alloys, and there are few studies on the effect of tin 56 content on the properties and properties of copper alloy disordered solid solutions." - The authors should refer these studies.

Lines 66-70: "Therefore, it is necessary to model Cu-Sn alloys with specific Sn content and study their 66 related properties. 67 In this paper, the phase stability, mechanical properties, and electronic properties of 68 Cu-Sn alloys with Sn content of 3.125 at%, 6.25 at% and 9.375 at% have been systematically 69 studied using a first principles calculation method." Why did you chose these specific values? Please, add your motivation.

2. Calculation method and details

"The calculations were all performed using the CASTEP" - Was the CASTEP code embedded in Materials Studio as you write here "and use the scripting modeling tool in Materials Studio"?

Lines 114-118: "As can be seen from 114 Table 2, the lattice constants of Cu31Sn, Cu30Sn2 and Cu29Sn3 deviate slightly from the fcc 115 structure; i.e., a ≠ b ≠ c, α ≠ β ≠ γ ≠ 90°. Since the symmetry of the crystal structure is slightly 116 broken after modeling the full relaxation optimization, this small deviation can be considered as an allowable error indicating the fcc structure." - So, after the relaxation, did you return the angles to 90 to use fcc symmetry operators?

Table 3/Figure 3. In introduction, you make this statement: "It is well 61 known that the solid solution limit of Sn in Cu matrix is 15.8 wt.% (9.2 at%) at a tempera- 62 ture of 500-600 ℃ [10]. When the Sn content increases to 15.8 wt.%, brittle intermetallic compounds such as δ- Cu41Sn11, which has a negative impact on ductility, greatly reduces the strength of the material, limiting its application and further development". However, based on your results given in Table 3 and Figure 3, the yield stress of 9.375 at.% Sn alloy is higher than for other alloys. So, you do not consider the formation of δ-Cu41Sn11 compound, right? In this case, what is your motivation to study this ratio? The results in this case seem to be incorrect.

Line 307."Where G is the shear modulus and b is the Bernoulli vector." - Did the authors mean Burgers vector?

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The authors should prove the validity of the results for the 9.375 at. % Sn alloy. If in reality brittle intermetallic compounds are formed in this alloy, then the obtained theoretical results are irrelevant and require a different approach to analysis.

The authors should check the grammar and some typos.

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"3.1. lattice constant" - with a capital letter.

Author Response

1. "However, currently, the calculation of copper tin alloys has been focused on the specific phase structure of copper alloys, and there are few studies on the effect of tin content on the properties and properties of copper alloy disordered solid solutions." - The authors should refer these studies.

Response 1: Thank you very much for your suggestion. We have amended the text accordingly: In the sentence "the calculation of copper tin alloys has been focused on the specific phase structure of copper alloys" is a summary of Ref. [7] [8]; "the effect of tin content on the properties and properties of copper alloy disordered solid solutions" cites Ref. [35].

2. Lines 66-70: "Therefore, it is necessary to model Cu-Sn alloys with specific Sn content and study their related properties. In this paper, the phase stability, mechanical properties, and electronic properties of Cu-Sn alloys with Sn content of 3.125 at%, 6.25 at% and 9.375 at% have been systematically studied using a first principles calculation method." Why did you choose these specific values? Please, add your motivation.

Response 2: We apologize for the ambiguity of the sentence due to our inaccurate expression. We have corrected the sentence to read: “Therefore, under the premise of ensuring the basic stability of the fcc structure, it is necessary to establish a model of Cu-Sn disordered solid solution to study the effect of tin solute on its related properties. In this study, the phase stability, mechanical properties, and electronic properties of Cu-Sn alloys with Sn content of 3.125 at%, 6.25 at% and 9.375 at% have been systematically studied using a first principles calculation method. The lattice constant, mixing enthalpy, yield stress, elastic constant, elastic modulus, density of state, differential charge density, and Debye temperature were calculated. This provides a theoretical basis for the subsequent research on Cu-Sn alloys and the design, development, and wide application of new copper alloys. It is worth mentioning that, according to the Cu-Sn phase diagram, the solid solution limit of Sn in Cu matrix is 15.8 wt.% (9.2 at%) [9], when the Sn content is greater than 15.8 wt.%, in addition to solid solution, δ-phase (Cu₄₁Sn₁₁) appears, which adversely affects the properties and applications of the material [10,11], and the generation of δ-phase should be avoided or reduced as much as possible in the practical production applications. Therefore, the δ-phase was not discussed in this study.” In Ref. [35], a similar discussion on the effect of Al content on the elastic properties of Cu-Al alloys has been made.

3. "The calculations were all performed using the CASTEP" - Was the CASTEP code embedded in Materials Studio as you write here "and use the scripting modeling tool in Materials Studio"?

Response 3: Thank you for pointing it that, as far as I have understood, for CASTEP, there are CASTEP for Linus system and CASTEP embedded in Materials studio (MS-CASTEP for short). The latter (MS-CASTEP) was used in this study.

4. Lines 114-118: "As can be seen from 114 Table 2, the lattice constants of Cu₃₁Sn, Cu₃₀Sn₂ and Cu₂₉Sn₃ deviate slightly from the fcc structure; i.e., a ≠ b ≠ c, α ≠ β ≠ γ ≠ 90°. Since the symmetry of the crystal structure is slightly 116 broken after modeling the full relaxation optimization, this small deviation can be considered as an allowable error indicating the fcc structure." - So, after the relaxation, did you return the angles to 90 to use fcc symmetry operators?

Response 4: Thank you very much for pointing it out. We have made corrections in the corresponding places in the manuscript. Through Ref. [19] [52], we have found that datas in Table 2 are indeed inappropriate, and the lattice constants are usually reserved to three or four decimal places, so we have corrected the data in Table 2 (reserved to three decimal places). The symmetry of the crystal structure is slightly broken after the structure optimization, but the fcc is still retained, so the fcc symmetry operators are not used to adjust the angles to 90° in the subsequent calculations.

5. Table 3/Figure 3. In introduction, you make this statement: "It is well 61 known that the solid solution limit of Sn in Cu matrix is 15.8 wt.% (9.2 at%) at a temperature of 500-600 ℃ [10]. When the Sn content increases to 15.8 wt.%, brittle intermetallic compounds such as δ- Cu₄₁Sn₁₁, which has a negative impact on ductility, greatly reduces the strength of the material, limiting its application and further development". However, based on your results given in Table 3 and Figure 3, the yield stress of 9.375 at% Sn alloy is higher than for other alloys. So, you do not consider the formation of δ-Cu₄₁Sn₁₁ compound, right? In this case, what is your motivation to study this ratio? The results in this case seem to be incorrect.

Response 5: Thank you very much for pointing it out. We have added the relevant description in the introduction, and as you said, according to the Cu-Sn phase diagram, when the Sn content is greater than 15.8 wt.% (9.2 at%), in addition to solid solution, δ-phase (Cu₄₁Sn₁₁) occurs, which negatively affects the material properties and applications, however, δ-phase should be avoided or reduced as much as possible in the practical production applications. Therefore, δ is not discussed in this study, which mainly focuses on systematically investigating the effect of the solid solution of Sn on the properties related to its Cu-Sn alloys. The Sn contents of 3.125 at%, 6.25 at% and 9.375 at% were selected to ensure that the fcc structure remains basically stable during modelling. In addition, observing the results given in Table 3 and Fig. 3, when the Sn content is 9.375 at%, the yield stress is higher than the other alloys, which may be due to other strengthening mechanisms such as solid solution strengthening, fine grain strengthening, etc., and not only by the δ-phase.

6. Line 307."Where G is the shear modulus and b is the Bernoulli vector." - Did the authors mean Burgers vector?

Response 6: I am very sorry, because of our carelessness, we mistakenly writed the “Burgers vector “as the ‘’Bernoulli vector“”. Corresponding modifications have also been made in this paper. Thank you very much!

7. The authors should prove the validity of the results for the 9.375 at. % Sn alloy. If in reality brittle intermetallic compounds are formed in this alloy, then the obtained theoretical results are irrelevant and require a different approach to analysis.

Response 7: Thank you very much for your suggestion, according to the Cu-Sn phase diagram, when the Sn content is greater than 9.2 at%, in addition to solid solution, the δ phase (Cu₄₁Sn₁₁) will occur, which negatively affects the material properties and applications, but the generation of δ phase should be avoided or minimized as much as possible in the practical production applications. Therefore, δ is not discussed in this study. In Ref. [35], a similar discussion on the effect of Al content on the elastic properties of Cu-Al alloys has been made.

8. The authors should check the grammar and some typos.

Response 8: Thank you for your kindness. We have corrected the sentence and checked the full text. Thank you very much!

9. "3.1. lattice constant" - with a capital letter.

Response 9: We are very sorry that due to our negligence, the writing is wrong. We have corrected "3.1. lattice constant" to "difference"3.1. Lattice constant".

Reviewer 2 Report

The Authors have written a very interesting paper about an important topic for many scientific and industrial applications. They have investigated in this paper the mechanism of the influence of Sn contents on the relevant properties of Cu-Sn alloys using DFT. It is especially important that the Authors have investigated the structure, elasticity, electronic and thermal properties of Cu-Sn alloys doped with different proportions of Sn (3.125 at%, 6.25 at% and 9.375 at%). The manuscript is interesting and clearly written, however with many grammatical errors. I would recommend publication after major revision.

 

1.       The Authors need to correct grammatical and English writing errors throughout the manuscript e.g. even in the Abstract the Authors write: “Firstly, its lattice constant and tin concentration comply with Vegas law.”, while it should be  Vegard's law, etc.

2.       Page 2, The Authors claim: “In this calculation, the Cu supercell containing 32 atoms with 2 × 2 × 2 expanded cells treated with fcc structure, refer to the script language written by Professor Zhao Yan, and use the scripting modeling tool in Materials Studio, using the method of atom random substitution, establish a Cu-Sn alloy stability models (Cu31Sn, Cu30Sn2, Cu29Sn3) doped with different proportions of Sn (where the mole fractions of Sn atoms are 3.125 at%, 6.250 88 at%, and 9.375 at%, respectively), as shown in Figure 1 and Table 1.” Can the Authors write more details about the script written by Professor Zhao Yan (or provide some references)? How does this method choose random structures? Can you provide details on how many random structures are generated in such a way per composition and how the Authors chose a representative model (is it the lowest total energy?)?

3.       Page 4, The Authors write in Table 2 Experimental and theoretical lattice parameters (a, b, c and α, β, γ) for Cu–Sn alloys. I would agree with the discussion on the previous page that a maximum error of  0.373% is a small one but the following text needs to be rephrased. I am not sure that the data in Table 2 can be computed with such accuracy (e.g. lattice parameter with 6 decimals?). If is still possible to have such precision within the code, then please explain how, otherwise, please round it to eg. 3 decimals. Also, all computed angles appear to be 90 degrees (if you round the lattice to 3 decimal points you will get cubic FCC which is also supported by later properties calculations). Please explain (and provide references if possible) how to get such tiny distortions or perhaps it is just within the accuracy of the CASTEP code and size of the supercell?

4.       Page 5, What is density flooding theory? Please explain.

5.       Page 6, The theoretical results of mismatch strain caused by Sn atoms in copper and the contribution of solution strengthening to the yield stress of Cu-Sn alloys presented in Table 3 differs a lot from the similar solid solutions strengthening experimental data. Can the Authors comment on that?

6.       Page 18. The Authors write: “3. By analyzing the electronic structure and bond characteristics, it is again shown that  Cu31Sn is the most stable and Cu30Sn2 has the highest Young's modulus and elastic anisotropy. Besides, Cu31Sn has strong phase stability and shear modulus, which can be attributed to the formation of strong Cu-Cu covalent bonds.” I am not sure what the Authors meant here since it is not easy to understand so please rephrase part 3 of the conclusion.

See above

Author Response

1. The Authors need to correct grammatical and English writing errors throughout the manuscript e.g. even in the Abstract the Authors write: “Firstly, its lattice constant and tin concentration comply with Vegas law.”, while it should be Vegard's law, etc.

Response 1: We are very sorry that due to our negligence, the writing is wrong. We have corrected "Vegas law" to "Vegard's law" and checked the entire manuscript.

2. Page 2, The Authors claim: “In this calculation, the Cu supercell containing 32 atoms with 2 × 2 × 2 expanded cells treated with fcc structure, refer to the script language written by Professor Zhao Yan, and use the scripting modeling tool in Materials Studio, using the method of atom random substitution, establish a Cu-Sn alloy stability models (Cu31Sn, Cu30Sn2, Cu29Sn3) doped with different proportions of Sn (where the mole fractions of Sn atoms are 3.125 at%, 6.250 88 at%, and 9.375 at%, respectively), as shown in Figure 1 and Table 1.” Can the Authors write more details about the script written by Professor Zhao Yan (or provide some references)? How does this method choose random structures? Can you provide details on how many random structures are generated in such a way per composition and how the Authors chose a representative model (is it the lowest total energy?)?

Response 2: Thank you very much for pointing it out. We have made corrections to the calculation methods and details in the manuscript, as follows: In this calculation, 2 × 2 × 2 supercells based on fcc structure were established by Perl Script enumeration of alloy structures. According to the Lowest Energy Principle, the stability models (Cu₃₁Sn, Cu₃₀Sn₂, and Cu₂₉Sn₃) of Cu-Sn alloys with Sn contents of 3. 125 at%, 6.250 at%, and 9.375 at% were screened out, respectively, as shown in Fig. 1 and Table 1. The detailed modelling methodology is shown in Appendix 1.

3. Page 4, The Authors write in Table 2 Experimental and theoretical lattice parameters (a, b, c and α, β, γ) for Cu–Sn alloys. I would agree with the discussion on the previous page that a maximum error of 0.373% is a small one but the following text needs to be rephrased. I am not sure that the data in Table 2 can be computed with such accuracy (e.g. lattice parameter with 6 decimals?). If is still possible to have such precision within the code, then please explain how, otherwise, please round it to e.g. 3 decimals. Also, all computed angles appear to be 90 degrees (if you round the lattice to 3 decimal points you will get cubic FCC which is also supported by later properties calculations). Please explain (and provide references if possible) how to get such tiny distortions or perhaps it is just within the accuracy of the CASTEP code and size of the supercell?

Response 3: Thank you very much for your suggestion, we have found that datas in Table 2 are indeed inappropriate through reference, and the lattice constants are generally retained to three to four decimal places, so we have corrected datas in Table 2 (retained to three decimal places), as you have said, all computed angles are all 90°, and the FCC of the Sn-doped elements is basically kept unchanged, which corresponds to the later performance calculations.

4. Page 5, What is density flooding theory? Please explain.

Response 4: We apologize for inadvertently writing: "density flooding theory" instead of "density functional theory", and we consider that it would be more appropriate to replace it with " based on first-principles", so we have made the correction in the manuscript.

5. Page 6, The theoretical results of mismatch strain caused by Sn atoms in copper and the contribution of solution strengthening to the yield stress of Cu-Sn alloys presented in Table 3 differs a lot from the similar solid solutions strengthening experimental data. Can the Authors comment on that?

Response 5: Thank you very much for pointing this out. Regarding the contribution of solid solution strengthening to the yield stress in Table 3 and Fig. 3, there is indeed a difference between the experimental results and the calculated results, and we believe that it is affected by a number of factors, for example, when the Sn content is the same, the yield strengths are different with different processing methods (the yield strength of the casting alloy is 139.2±16.6 MPa when the Sn content is 8.688 at%, while that of the SLM alloy strength is 436±3 MPa); in addition, when both Sn content and processing method are the same, the yield strength is different for different treatment conditions (the yield strength of the SLM alloy is 436±3 MPa; while the yield strength of the SLM+annealing alloy is 328±4 MPa). Since the first-principles used in this paper were carried out at 0 K, which is different from the experimental conditions, the yield strength values are different, but the general trend remains the same.

6. Page 18. The Authors write: “3. By analyzing the electronic structure and bond characteristics, it is again shown that Cu₃₁Sn is the most stable and Cu₃₀Sn₂ has the highest Young's modulus and elastic anisotropy. Besides, Cu₃₁Sn has strong phase stability and shear modulus, which can be attributed to the formation of strong Cu-Cu covalent bonds.” I am not sure what the Authors meant here since it is not easy to understand so please rephrase part 3 of the conclusion.

Response 6: Thank you very much for pointing it out. We apologize for the ambiguity in the meaning of the sentence due to some inappropriate descriptions. The conclusion of the manuscript was obtained from t Ref. [52] as well as from the analysis comparing the bonding properties in Table 6 with the stability in Table 3 and the elastic properties in Table 5. It can be found that the changes in phase stability and shear modulus of Cu-Sn alloys in this study are positively correlated with the strength of the Cu-Cu covalent bond. We have now made a correction in the manuscript: The electronic structure and bonding properties of the Cu-Sn alloys have been calculated, and their relationship with the stability and mechanical properties of the alloys is analyzed and discussed. Three types of bonding existed in Cu-Sn alloys: Cu-Cu covalent bonds, Cu-Cu metallic bonds, and Cu-Sn covalent bonds, of which Cu₃₁Sn had the best stability and the highest shear modulus, which depended to a certain extent on the fact that it had stronger Cu-Cu covalent bonds.

Reviewer 3 Report

The Cu-Sn alloys has a number of important applications in  aerospace, marine and consumer electrical devices. Thus, detailed researches of practical properties  of the materials such as wear and corrosion resistance, thermal and electrical conductivity are of real importance. The paper under review is a step in this direction.

My doubts are related with Fig.7 of the article. Where is the conduction zone of Cu? In Fig.7,  looking at the PDOS of Cu, one can see the conduction zone is very weakly expressed (in fact, it is absent). Electrical conductivity of the alloy is only due to the impurity of Sn atoms. Are the authors sure of the result ? A detailed explanation from their side is absolutely needed.

Author Response

1. My doubts are related with Fig.7 of the article. Where is the conduction zone of Cu? In Fig.7, looking at the PDOS of Cu, one can see the conduction zone is very weakly expressed (in fact, it is absent). Electrical conductivity of the alloy is only due to the impurity of Sn atoms. Are the authors sure of the result? A detailed explanation from their side is absolutely needed.

 Response 1: Thank you very much for pointing it out. We apologize for the ambiguity in the meaning of the sentence due to some inappropriate descriptions. We have made corrections in the corresponding parts of the text: First, the TDOS below the Fermi energy level (0 eV) is contribution mainly by the Cu-3d states with partial contributions from the Sn-5s and Sn-5p states, while the TDOS above the Fermi energy level mainly originates from the Sn-5s and Sn-5p states, while partly from the Cu-3p states.  As shown in Fig. 7 (currently corrected to Fig. 6), the s and p orbitals of Sn are the main sources of the conduction band, and the p orbital of Cu has a small contribution. By analyzing and comparing the results of the density of states and fractional density of states of Cu₃Sn intermetallic compounds in Ref. [47]. we believe that the result of the present work is reliable.

Reviewer 4 Report

The manuscript reports a quite extensive study of various properties of Cu-Sn alloys for a few concentrations of Sn, based on Density Functional Theory formalism. The results are, in general, valuable and of interest to the community and cover a wide spectrum of properties. However, before recommending acceptance of the manuscript by the journal Crystals, I would like the Authors to address the points listed below:

* Line 39: The statement “based on quantum mechanics” does not seem necessary.

* Line 83: “exchange-correlation energy” would sound better than “generalization”. Also, “structure energy and stability” is not general enough; “on the results should be enough.

* Line 85: The statement “refer to the script language written by Professor Zhao Yan” is unclear; if the script was published, please cite the referring web page or other source. If not, I believe that this information it is of little value to the Reader. Also the whole description of the used cell is a little bit unclear: is it just a 2x2x2 supercell based on fcc structure? This would be explained simpler.

* The part of the sentence in lines 87-89 is too long and unclear. Please split it into two sentences and correct generally.

* Line 111: I guess that the difference would be just called ‘difference’, not ‘error’ and it is not maximum error, as just two values are compared.

* Line 113: ‘better’ would be replaced with ‘good’ or other word; there is no comparison in the sentence.

* Equation 1: the information that R^2=0 is not necessary in equation. More precise value of R is given in Fig. 2.

 

* Regarding the data in Table 2 and their discussion: I am unsure if the obtained differences between a, b and c are really physically meaningful in the light of the precision of the calculations. Also the deviations of the angles from 90 deg. seem below the overall accuracy. If yes, then just the average values of obtained a,b,c would be given as a, not enumerating separately b and c (and not enumerating the angles).

 

* Fig. 2: Please correct the caption. Also, can the linear function be fitted also to the experimental data to strengthen the discussion?

* Line 128: Regarding the comment on thermal expansion – what was the temperature for which the experimental data on a were measured according to the cited literature? This information would be included in the Table 2.

* Please try to explain more clearly the reason for using Birch-Murnaghan equation in the procedure of determination of the mixing enthalpy. If any fitting was done to this equation, maybe the results would be given in some appendix? Please try to describe the procedure a little bit more.

* Regarding Table 3: please explain in more detail why the value of epsilon is an average for all concentrations? It seems unclear.

* Lines 272-3: please correct the sentence, it sounds unclear.

 

* Figure 4: please mark in clear way what symbols correspond to the results of the present calculation.

 

* The citations of some references lack important bibliographic data – this should be completed – see for example Ref. 36, 39, 54, 59, 61.

* General remark: some language polishing would be necessary. Also some mistakes (like the one in line 16: ‘Vegard’s’ instead of ‘Vegas’) should be corrected.

Author Response

1. Line 39: The statement “based on quantum mechanics” does not seem necessary.

Response 1: Thanks to your suggestion, we have made a correction in the manuscript by deleting "based on quantum mechanics" at the appropriate place.

2. Line 83: “exchange-correlation energy” would sound better than “generalization”. Also, “structure energy and stability” is not general enough; “on the results should be enough.

Response 2: Thank you very much for pointing it out, we have changed "exchange-correlation generalization" to "exchange-correlation energy" and generalised "structure energy and stability" to "results" in the manuscript.

3. Line 85: The statement “refer to the script language written by Professor Zhao Yan” is unclear; if the script was published, please cite the referring web page or other source. If not, I believe that this information it is of little value to the Reader. Also the whole description of the used cell is a little bit unclear: is it just a 2x2x2 supercell based on fcc structure? This would be explained simpler.

Response 3: Thank you very much for your suggestion. We have made corrections to the calculation methods and details in the manuscript, as follows: In this calculation, 2 × 2 × 2 supercells based on fcc structure were established by Perl Script enumeration of alloy structures. According to the Lowest Energy Principle, the stability models (Cu₃₁Sn, Cu₃₀Sn₂, and Cu₂₉Sn₃) of Cu-Sn alloys with Sn contents of 3. 125 at%, 6.250 at%, and 9.375 at% were screened out, respectively, as shown in Fig. 1 and Table 1. The detailed modelling methodology is shown in Appendix 1.

4. The part of the sentence in lines 87-89 is too long and unclear. Please split it into two sentences and correct generally.

Response 4: Thank you very much for pointing it out. We have made the correction at the appropriate place in the manuscript: According to the Lowest Energy Principle, the stability models (Cu₃₁Sn, Cu₃₀Sn₂, and Cu₂₉Sn₃) of Cu-Sn alloys with Sn contents of 3. 125 at%, 6.250 at%, and 9.375 at% were screened out, respectively, as shown in Fig. 1 and Table 1. The detailed modelling methodology is shown in Appendix 1.

5. Line 111: I guess that the difference would be just called ‘difference’, not ‘error’ and it is not maximum error, as just two values are compared.

Response 5: Thank you very much for your suggestion, we have corrected "maximum error" to "difference" in the manuscript.

6. Line 113: ‘better’ would be replaced with ‘good’ or other word; there is no comparison in the sentence.

Response 6: Thank you very much for your suggestion, we have corrected "better" to "good" in the manuscript.

7. Equation 1: the information that R^2=0 is not necessary in equation. More precise value of R is given in Fig. 2.

 Response 7: Thanks to your suggestion, we have made a correction at the appropriate place in the manuscript: remove "with R² = 0".

8. Regarding the data in Table 2 and their discussion: I am unsure if the obtained differences between a, b and c are really physically meaningful in the light of the precision of the calculations. Also the deviations of the angles from 90 deg. seem below the overall accuracy. If yes, then just the average values of obtained a,b,c would be given as a, not enumerating separately b and c (and not enumerating the angles).

 Response 8: Thank you very much for pointing it out. We have made corrections in the corresponding places in the manuscript. Through Ref. [19] [52], we have found that the data in Table 2 are indeed inappropriate, and the lattice constants are usually reserved to three or four decimal places, so we have corrected the data in Table 2 (reserved to three decimal places).

9. Fig. 2: Please correct the caption. Also, can the linear function be fitted also to the experimental data to strengthen the discussion?

 Response 9: Thank you very much for your suggestion. We have corrected the caption of Fig. 2 from "Chang of lattice constant with Sn concentration in Cu-Sn alloys." to "Relationship between the lattice parameter and the atomic Sn concentration of Cu-Sn alloys", as shown in Fig.2. In addition, the experimental values have been linearly fitted, The relevant equation is as follows:

a (Å) = 3.615 + 1.054c       with   R=0.9997      

Figure 2. Relationship between the lattice parameter and the atomic Sn concentration of Cu-Sn alloys.

10. Line 128: Regarding the comment on thermal expansion – what was the temperature for which the experimental data on a were measured according to the cited literature? This information would be included in the Table 2.

Response 10: Thank you very much for your suggestions. We have made the addition in Table 2: The experimental data of a was performed at 25°C.

11. Please try to explain more clearly the reason for using Birch-Murnaghan equation in the procedure of determination of the mixing enthalpy. If any fitting was done to this equation, maybe the results would be given in some appendix? Please try to describe the procedure a little bit more.

Response 11: Thank you very much for pointing it out. We have added relevant information in the corresponding position in the manuscript: “The Birch-Murnaghan equation reveals the internal structure and properties of solids by investigating their rate of change of volume and modulus of elasticity at different pressures. The energy-volume (E-V) curve can be obtained by fitting this equation to obtain the total static energy of the pure element, which can then be substituted to obtain the enthalpy of mixing.” and “ The specific calculated data (equilibrium volume V₀（/atom）, bulk modulus B₀ （）, first-order derivative of bulk modulus with respect to pressure B₀’ and static energy E₀（/atom）of Cu and Sn.) and the fitted E-V curves of Cu and Sn are shown in Appendix 2 ,Appendix 3,and Volume V（/atom）and total energy E（/atom）of the Cu–Sn alloys are shown in Appendix 4.”

12. Regarding Table 3: please explain in more detail why the value of epsilon is an average for all concentrations? It seems unclear.

Response 12: Thank you very much for pointing out that the values of the misfit strains ε in Table 3 being taken as an average of the different concentrations, we made the decision based on Ref. [20], which we have labelled in the manuscript. The literature states that "These misfit strains of Al-X solid solution are based on the average values of the first nearest-neighbor interatomic distances in Al₂₆X₁, Al₃₁X₁, Al₆₃X₁, and Al₁₀₇X₁."

13. Lines 272-3: please correct the sentence, it sounds unclear.

 Response 13: Thank you for pointing it out. We have corrected the sentence in the manuscript at the appropriate places: Calculated values of elastic parameters for Cu and Cu-Sn alloys (Cu₃₁Sn, Cu₃₀Sn₂ and Cu₂₉Sn₃) are presented in Table 5. In order to verify the reliability of the calculated results, the calculated values of the elastic parameters for copper in Table 4 and Table 5 were compared with the previously reported experimental values [40-42] and theoretical values [43,44]. It can be seen that the elastic parameters obtained in this study are in better agreement with the reference values, indicating that the calculated parameters and method have high reliability and certain reference value.

14. Figure 4: please mark in clear way what symbols correspond to the results of the present calculation.

 Response 14: Thank you very much for pointing this out. After checking, we found that the content of Figure 4 has been clearly indicated in Tables 4 and 5, so Figure 4 has been deleted from the manuscript.

15. The citations of some references lack important bibliographic data – this should be completed – see for example Ref. 36, 39, 54, 59, 61.

Response 15: Thank you for pointing it out. We have revised the references according to journal's requirements. It is worth noting that the order of references in the text has changed (e.g., the original Ref. 36, 39, 54, 59, 61 have been corrected to the current Ref. 35, 38, 53, 58, 60). Details are as follows :

Original reference: 36. A. Jy.;A. Po. First-Principles Study of the Effect of Aluminum Content on the Elastic Properties of Cu-Al Alloys. 2022.

Revised reference: 35. A. Jy.;A. Po. First-Principles Study of the Effect of Aluminum Content on the Elastic Properties of Cu-Al Alloys.  Materials Today Communications 2022, 31, 103399.

Original reference: 39. I. Waller. Dynamical Theory of Crystal Lattices by M. Born and K. Huang.  Acta Crystallographica 1956, 9.

Revised reference: 38. I. Waller. Dynamical Theory of Crystal Lattices by M. Born and K. Huang.  Acta Crystallographica 1956, 9, 837-838.

Original reference: 54. Y.A. Wei.;A. Yz. Investigation on elastic properties and electronic structure of dilute Ir-based alloys by first-principles calculations.  Journal of Alloys and Compounds 850.

Revised reference: 53. Y.A. Wei.;A. Yz. Investigation on elastic properties and electronic structure of dilute Ir-based alloys by first-principles calculations.  Journal of Alloys and Compounds 2021, 850, 156548.

Original reference: 59. E. Schreiber.;Author. Elastic Constants and Their Measurement.  Journal of Applied Mechanics 1975.

Revised reference: 58. E. Schreiber.;Author. Elastic Constants and Their Measurement.  Journal of Applied Mechanics 1975, 42, 747-748.

Original reference: 61. E. M..;Fine. Elastic constants versus melting temperature in metals.  Scripta Metallurgica 1984,

Revised reference: 60. E. M..;Fine. Elastic constants versus melting temperature in metals.  Scripta Metallurgica 1984, 18, 951-956.

16. General remark: some language polishing would be necessary. Also some mistakes (like the one in line 16: ‘Vegard’s’ instead of ‘Vegas’) should be corrected.

Response 16: We are very sorry that due to our negligence, the writing order is wrong. We have corrected "Vegas law" to "Vegard's law" and checked the entire manuscript.

Round 2

Reviewer 1 Report

The authors have made all the necessary corrections. The article can be published in the current form. 

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Author Response

Thank you again for your suggestions and for agreeing with our response. In the meanwhile, we have checked the full manuscript and made further corrections to the language of the manuscript.