Mixed-Mode Fracture Behavior of Penta-Graphene: A Molecular Dynamics Perspective on Defect Sensitivity and Crack Evolution
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThe manuscript employs molecular dynamics simulations to investigate the fracture behavior of penta-graphene, a novel two-dimensional material, with a particular focus on the interplay between structural defects and complex loading conditions. The results offer valuable theoretical insights for the future design and failure prediction of nanomaterials. However, the manuscript contains several issues and is not recommended for publication in its current form. The detailed comments are as follows:
- The manuscript is a bit wordy. For example, Figure 1 is discussed extensively and repetitively in both Section 1 and Section 2. The potential applications and research significance of the material are also emphasized multiple times across different sections, which diminishes the overall readability. The authors are advised to streamline the text to improve clarity and make the manuscript easier to follow.
- In Section 2, the atomic structure of penta-graphene is described, including lattice constants, bond lengths, and the ratio of sp² to sp³ The authors should provide appropriate references for the modeling details and clarify the sources of these parameters.
- In Section 3, the system is relaxed using the NPT ensemble from low to room temperature, followed by NVE relaxation. This raises a concern: since the subsequent tensile simulations are also performed under the NPT ensemble, is the NVE relaxation step necessary?
- In Figure 2b, the curves corresponding to 10%, 15%, and 20% do not follow the same trend as the authors’ conclusion, nor are they consistent with the trends shown in Figures 2a, 2c, and 2d. Please provide a reasonable explanation for this inconsistency.
- In Section 4.2, the authors mention that “The cracks introduced into the pristine material range from 30.59Å to 73.78Å in width, increasing in increments of 7.2Å.”, yet in subsequent text “The stress distribution and crack growth at different strain levels for a penta-graphene sheet with an initial crack length of 52.18 Å are shown in Fig. 8.”, it is referred to as crack length. Please check and ensure the accuracy of these descriptions.
- In the analysis of Figure 11, the authors state that the elastic modulus is minimized at 45°, yet the figure clearly shows that the elastic modulus at 45° is not the lowest. Besides, the fracture strength of 45° is even the highest. Please clarify and correct this statement.
Author Response
Response to Reviewer 1 Comments
Thank you very much for taking the time to review this manuscript. Please find the detailed responses below and the corresponding revisions/corrections highlighted/in track changes in the re-submitted files.
Comments 1: The manuscript is a bit wordy. For example, Figure 1 is discussed extensively and repetitively in both Section 1 and Section 2. The potential applications and research significance of the material are also emphasized multiple times across different sections, which diminishes the overall readability. The authors are advised to streamline the text to improve clarity and make the manuscript easier to follow.
Response 1: We have modified sections 1 and 2. The description of Figure 1 is removed from section 1, and it is only included in section 2. Also, repeated lines regarding the potential application and research significance of the material are removed from section 2.
Comments 2: In Section 2, the atomic structure of penta-graphene is described, including lattice constants, bond lengths, and the ratio of sp² to sp³ the authors should provide appropriate references for the modeling details and clarify the sources of these parameters.
Response 2: We have added two references for the atomic structure of penta-graphene in section 2. The references are on page 4, reference 22 and 37.
Comments 3: In Section 3, the system is relaxed using the NPT ensemble from low to room temperature, followed by NVE relaxation. This raises a concern: since the subsequent tensile simulations are also performed under the NPT ensemble, is the NVE relaxation step necessary?
Response 3: The NVE relaxation step is necessary to stabilize the total energy of the system. We have mentioned that on page 5, lines 184-186.
Comments 4: In Figure 2b, the curves corresponding to 10%, 15%, and 20% do not follow the same trend as the authors’ conclusion, nor are they consistent with the trends shown in Figures 2a, 2c, and 2d. Please provide a reasonable explanation for this inconsistency.
Response 4: An inconsistency in the trends is not unexpected, as each simulation is run with a different random generation of point defects. The interaction between defects can cause some simulations to fracture earlier as local stress concentrations develop. The slight inconsistency does not diminish the overall trend of decreasing Young’s modulus and fracture strength with increasing void defect percentages.
Comments 5: In Section 4.2, the authors mention that “The cracks introduced into the pristine material range from 30.59Å to 73.78Å in width, increasing in increments of 7.2Å.”, yet in subsequent text “The stress distribution and crack growth at different strain levels for a penta-graphene sheet with an initial crack length of 52.18 Å are shown in Fig. 8.”, it is referred to as crack length. Please check and ensure the accuracy of these descriptions.
Response 5: The results shown in Figure 8 are only for the initial crack length of 52.18 Å, which is the middle length of the 7 we assessed. The x-axis of Figure 6 shows which crack lengths. The 7.2 Å increment is a rounded value for twice the unit cell width. Each new crack length is 2 unit cells wider than the previous crack. This is chosen to keep the atomic layout of each crack tip identical.
Comments 6: In the analysis of Figure 11, the authors state that the elastic modulus is minimized at 45°, yet the figure clearly shows that the elastic modulus at 45° is not the lowest. Besides, the fracture strength of 45° is even the highest. Please clarify and correct this statement.
Response 6: We have corrected the text for the related paragraphs in Section 4.3.
Reviewer 2 Report
Comments and Suggestions for AuthorsThis paper presents an extensive molecular-dynamics study of the mechanical and fracture behavior of penta-graphene, focusing on defect sensitivity, loading direction, and mixed-mode fracture toughness. The simulations are carefully performed and the results (stress–strain curves, crack-propagation pathways, critical stress-intensity factors) are presented clearly. But several key points need further addressing before publication - please see below.
1. Introduction – Broader Property Comparison
The authors compare band gaps in graphene vs penta-graphene but only qualitatively summarize mechanical, electrical, and thermal properties. Please provide a more quantitative side-by-side comparison (e.g. Young’s modulus, thermal conductivity, carrier mobility) to make clear where penta-graphene stands relative to graphene in all three domains .
Relatedly, other methods (strain engineering, chemical functionalization) can open a band gap in graphene. The manuscript should discuss the advantages of using an intrinsically gapped allotrope versus, for example, anisotropic strain, and whether penta-graphene offers superior electronic performance or stability.
2. Scalability and Synthesis
The paper notes that penta-graphene “remains poorly understood” experimentally due to synthesis challenges . Please add a discussion of potential synthetic routes or scalable fabrication strategies and comment on the expected defect levels in realistic samples.
3. Uniqueness and Metastability of the Structure
The atomic-structure section details the ideal unit cell but does not explore whether other low-energy tilings or reconstructions exist. Can the authors enumerate how many meta-stable configurations arise if one perturbs the pure Cairo tiling? What are the energy barriers to transition between them? This is important to assess whether the chosen structure is kinetically accessible .
4. Interatomic Potential Validation and Sensitivity
While the Erhart–Albe Tersoff potential is well justified by prior benchmarking, please test at least one additional potential (e.g., a REBO or AIREBO variant) on a representative set of properties (elastic modulus, fracture strength) to confirm the robustness of key trends.
More generally, include a sensitivity analysis showing how quantitative results (e.g., critical SIFs, elastic constants) vary with simulation parameters: strain rate, thermostat/barostat settings, system size.
5. Strain-Application Protocol
The Methods apply a continuous strain rate of 0.001 ps⁻¹. Please justify this choice: how would an instantaneous (step) strain or a lower/higher rate affect the observed fracture mechanisms? A brief discussion will help readers assess the realism of the loading protocol .
6. Loading-Direction Study and Structural Variants
The anisotropy results (modulus vs rotation angle) are convincing, but it remains unclear how unique they are to the ideal pentagonal tiling. Could slight perturbations (e.g. introducing small out-of-plane ripples or rotated domains) change the angle of minimum stiffness? A comment on the generality of the anisotropy trends would strengthen this section .
Overall I find the study comprehensive and of interest, but I recommend major revision to address the points above. Once the authors have supplied the additional comparisons, validation tests, and clarifications, the manuscript will be significantly stronger.
Author Response
Response to Reviewer 2 Comments
Thank you very much for taking the time to review this manuscript. Please find detailed responses below and the corresponding revisions/corrections highlighted/in track changes in the re-submitted files.
Comments 1: Introduction – Broader Property Comparison
The authors compare band gaps in graphene vs penta-graphene but only qualitatively summarize mechanical, electrical, and thermal properties. Please provide a more quantitative side-by-side comparison (e.g. Young’s modulus, thermal conductivity, carrier mobility) to make clear where penta-graphene stands relative to graphene in all three domains.
Relatedly, other methods (strain engineering, chemical functionalization) can open a band gap in graphene. The manuscript should discuss the advantages of using an intrinsically gapped allotrope versus, for example, anisotropic strain, and whether penta-graphene offers superior electronic performance or stability.
Response 1: The quantitative comparison between graphene and penta-graphene in terms of band gaps, mechanical, electrical, and thermal properties have been included in section 1. The comparison can be found on page 3, lines 71 to 78..
Comments 2: Scalability and Synthesis
The paper notes that penta-graphene “remains poorly understood” experimentally due to synthesis challenges . Please add a discussion of potential synthetic routes or scalable fabrication strategies and comment on the expected defect levels in realistic samples.
Response 2: Two possible synthesis routes of penta-graphene have been included in the Introduction, page 2 and 3 (Lines 53 to 70).
Comments 3: Uniqueness and Metastability of the Structure
The atomic-structure section details the ideal unit cell but does not explore whether other low-energy tilings or reconstructions exist. Can the authors enumerate how many meta-stable configurations arise if one perturbs the pure Cairo tiling? What are the energy barriers to transition between them? This is important to assess whether the chosen structure is kinetically accessible.
Response 3: We agree that an investigation into metastable configurations and energy barriers associated with perturbations of the Cairo tiling would provide a deeper understanding of penta-graphene structure. However, this analysis involves extensive first-principles or advanced sampling techniques, which fall beyond the scope of our current molecular dynamics-based mechanical study. Our focus here is to evaluate the fracture and mechanical behavior of the penta-graphene configuration.
Comments 4: Interatomic Potential Validation and Sensitivity
While the Erhart–Albe Tersoff potential is well justified by prior benchmarking, please test at least one additional potential (e.g., a REBO or AIREBO variant) on a representative set of properties (elastic modulus, fracture strength) to confirm the robustness of key trends.
More generally, include a sensitivity analysis showing how quantitative results (e.g., critical SIFs, elastic constants) vary with simulation parameters: strain rate, thermostat/barostat settings, system size.
Response 4: We performed a simulation using the AIREBO potential on a pristine penta-graphene sheet (120 Å in length and 60 Å in width), free of any initial defects. The mechanical test using AIREBO produced a yield strength of 48.936 N/m, which aligns well with values reported in recent literature (Bedi et al., 2024). The corresponding stress-strain curve is presented in Figure 1.
However, the stress–strain curve obtained using the AIREBO potential exhibited a convex shape during the elastic regime, consistent with the findings of Bedi et al. (2024). This behavior is atypical for most materials under tensile loading, where a concave or linear response is expected in the elastic range. The presence of a convex curve suggests that AIREBO may not accurately capture the fundamental mechanical response of penta-graphene, particularly in the linear-elastic regime. As a result, we adopted the Tersoff potential developed by Erhart and Albe, which has been extensively validated in prior studies. Winczewski et al. (2018), for instance, conducted a rigorous benchmarking analysis of 14 different empirical potentials—including Tersoff, Stillinger-Weber, REBO, and REAX variants—and concluded that only the Erhart–Albe Tersoff potential accurately replicates the mechanical and structural characteristics of penta-graphene.
In our simulations, the Tersoff potential produced a stress-strain response with a clear linear elastic region and a physically consistent failure profile. These results reinforce its appropriateness for simulating the fracture behavior of penta-graphene and support our decision to use this potential throughout the study to ensure predictive reliability.
Figure: 1
References:
Bedi, D.; Sharma, S.; Tiwari, S. Exploring the novel carbon allotropes: Phagraphene and Pentagraphene. Materials Today: Proceedings 2024.
Winczewski, S.; Shaheen, M.Y.; Rybicki, J. Interatomic potential suitable for the modeling of penta-graphene: Molecular statics/molecular dynamics studies. Carbon 2018, 126, 165–175
Comments 5: Strain-Application Protocol
The Methods apply a continuous strain rate of 0.001 ps⁻¹. Please justify this choice: how would an instantaneous (step) strain or a lower/higher rate affect the observed fracture mechanisms? A brief discussion will help readers assess the realism of the loading protocol .
Response 5: A discussion of the chosen strain rate as it relates to other studies has been added to Section 3, page 5 , lines 189 to 192.
A discussion of the fracture mechanism’s dependence on strain rate has been added to Section 4.2.
Comments 6: Loading-Direction Study and Structural Variants
The anisotropy results (modulus vs rotation angle) are convincing, but it remains unclear how unique they are to the ideal pentagonal tiling. Could slight perturbations (e.g. introducing small out-of-plane ripples or rotated domains) change the angle of minimum stiffness? A comment on the generality of the anisotropy trends would strengthen this section.
Overall I find the study comprehensive and of interest, but I recommend major revision to address the points above. Once the authors have supplied additional comparisons, validation tests, and clarifications, the manuscript will be significantly stronger.
Response 6: We’ve included a brief discussion of the expected generality of the anisotropic behavior in section 4.3.
Reviewer 3 Report
Comments and Suggestions for AuthorsThis manuscript describes Mixed-Mode Fracture Behavior of Penta-Graphene with A Molecular Dynamics Perspective on Defect Sensitivity and Crack Evolution.
The 21-page manuscript is divided into different paragraphs. It is well-written, easy to read, and clear.
After carefully reading the manuscript, one main question arises.
The third paragraph emphasizes the method using LAMMPS and the Tersoff potential, as justified by the publication of Winczewski et al. (ref. 45).
As with DFT, where functionals are selected according to the targets, the use of the Tersoff potential should be validated, or at least compared with another potential, in the case of voids and fractures.
It can be done with the results reported in Figure 2. In my opinion, it is necessary to test an alternative potential, such as REBO2 or AIREBO, to determine if the same results are obtained, meaning that voids are adequately managed, as Tersoff is not generally considered optimal for describing edges and voids.
Author Response
Response to Reviewer 3 Comments
Thank you very much for taking the time to review this manuscript. Please find detailed responses below and the corresponding revisions/corrections highlighted/in track changes in the re-submitted files.
Comments 1: The third paragraph emphasizes the method using LAMMPS and the Tersoff potential, as justified by the publication of Winczewski et al. (ref. 45).
As with DFT, where functionals are selected according to the targets, the use of the Tersoff potential should be validated, or at least compared with another potential, in the case of voids and fractures.
It can be done with the results reported in Figure 2. In my opinion, it is necessary to test an alternative potential, such as REBO2 or AIREBO, to determine if the same results are obtained, meaning that voids are adequately managed, as Tersoff is not generally considered optimal for describing edges and voids.
Response 1: We performed a simulation using the AIREBO potential on a pristine penta-graphene sheet (120 Å in length and 60 Å in width), free of any initial defects. The mechanical test using AIREBO produced a yield strength of 48.936 N/m, which aligns well with values reported in recent literature (Bedi et al., 2024). The corresponding stress-strain curve is presented in Figure 1.
However, the stress–strain curve obtained using the AIREBO potential exhibited a convex shape during the elastic regime, consistent with the findings of Bedi et al. (2024). This behavior is atypical for most materials under tensile loading, where a concave or linear response is expected in the elastic range. The presence of a convex curve suggests that AIREBO may not accurately capture the fundamental mechanical response of penta-graphene, particularly in the linear-elastic regime. As a result, we adopted the Tersoff potential developed by Erhart and Albe, which has been extensively validated in prior studies. Winczewski et al. (2018), for instance, conducted a rigorous benchmarking analysis of 14 different empirical potentials—including Tersoff, Stillinger-Weber, REBO, and REAX variants—and concluded that only the Erhart–Albe Tersoff potential accurately replicates the mechanical and structural characteristics of penta-graphene.
In our simulations, the Tersoff potential produced a stress-strain response with a clear linear elastic region and a physically consistent failure profile. These results reinforce its appropriateness for simulating the fracture behavior of penta-graphene and support our decision to use this potential throughout the study to ensure predictive reliability.
Figure: 1
References:
Bedi, D.; Sharma, S.; Tiwari, S. Exploring the novel carbon allotropes: Phagraphene and Pentagraphene. Materials Today: Proceedings 2024.
Winczewski, S.; Shaheen, M.Y.; Rybicki, J. Interatomic potential suitable for the modeling of penta-graphene: Molecular statics/molecular dynamics studies. Carbon 2018, 126, 165–175.
Round 2
Reviewer 1 Report
Comments and Suggestions for AuthorsThe revised manuscript is acceptable. I have no further comments.
Reviewer 2 Report
Comments and Suggestions for AuthorsMost of the previous comments have been adequately addressed. I recommend the publication of this manuscript.
