It is our pleasure to submit the manuscript entitled “Bimodal grain-size scaling of thermal transport inpolycrystalline graphene from large-scale molecular dynamics simulations” by Zheyong Fan, Petri Hirvonen, Luiz Felipe C. Pereira, Mikko M. Ervasti, Ken R.Elder, Davide Donadio, Ari Harju, and Tapio Ala-Nissila, for exclusive publication in Nano Letters.
Graphene holds great potential for thermal management applications due to its exceptionally high thermal conductivity (which can exceed 5 000 W/mK) in its pristine form. However, wafer-scale graphene samples needed for industrial applications are usually grown by chemical vapor deposition and are inevitably polycrystalline in nature. Such samples contain grain boundaries which are extended line defects separating grains of different orientations. It has been known since the seminal 1941 experiments by P. Kapitza in liquid He that interfaces have amajor impact on the heat flow across them. While grain boundary and other interface effects on heat conduction have been extensively studied in 3D systems, their influence on 2D materials such as graphene is much less understood.
The central question for polycrystalline graphene concerns the scaling of the thermal conductivity with the average grain size. To this end, in our work we employ extensive classical molecular dynamics (MD) simulations to quantify the grain-size scaling of thermal transport in large suspended samples prepared by an efficient multiscale approach based on the phase field crystal model, which allows us to consider properly thermalized multigrain configurations up to about 200 nm in linear size.
Our first important result is that in contrast to previous theoretical works, the scaling of the thermal conductivity with the grain size displays bimodal behaviour with two effective Kapitza lengths. Compared to pristine graphene where the thermal conductivity is dominated by flexural modes associated with out-of-plane phonons, in polycrystalline graphene the grain size scaling is dominated by the out-of-plane (flexural) phonons with a Kapitza length that is an order of magnitude larger than that of the in-plane phonons.This result quantifies for the first time the dramatic influence of grain boundaries on heat conduction in 2D materials.
Concerning heat conduction measurements in graphene there is a large variation between results from different experiments. Most recently it has been shown by high-quality experiments that in pristine graphene the heat conductivity can be as high as 5 000 W/mK (as mentioned above). This value is almost 70% higher than that obtained from the most accurate classical MD simulations. This means that quantum effects play an important role. The second important result in our work is that we show that when mode-by-mode quantum corrections are properly applied to our classical MD results (and the pristine case is properly renormalized to the experimental value), we can obtain full agreement with the experimental data for polycrystalline graphene.
To summarize, we think that our results constitute a significant step forward in understanding heat transport in graphene and other two-dimensional materials. Our work should be of broad interest to engineers, chemists and physicists working on nano-structured graphene and related systems in the context of thermal management devices and deserves publication in Nano Letters.
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Thank you for a rapid processing of our manuscript exclusively submitted for publication in Nano Letters. We hereby submit a revised version of the manuscript “Bimodal grain-size scaling of thermal transport in polycrystalline graphene from large-scale molecular dynamics simulations” by Zheyong Fan, Petri Hirvonen, Luiz Felipe C. Pereira, Mikko Ervasti, Ken Elder, Davide Donadio, Ari Harju, and Tapio Ala-Nissila (Manuscript ID: nl-2017-01742z). We thank the reviewers for their constructive criticism that has helped us to improve themanuscript. Since the changes to the manuscript are relatively minor we hope that our work can now be accepted for publication.
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In this manuscript, the authors present a convincing study on thermal conductivity of polycrystalline graphene using molecular dynamics calculations. The novelty of the work is based on the (more than usual) realistic model for samples as well as on the quantum corrections applied on the classical simulations. In my view the article is suitable for publication in Nano Letters as is.
Reply: We thank the referee for supporting our work to be published in Nano Letters.
The authors perform large-scale quantum-corrected MD simulations of polycrystalline graphene. The results are compared to one set of recent experiments and used to explain different gran size scaling trends for in-plane and out-of-plane phonons. The work is quite interesting because it is among the first MD results that accurately predicts Kaptiza resistances at grain boundaries, and as such may be of broad interest. However, there are several questions that arise:
Reply:We thank the Referee for the positive comments.
1) Ref. 16 clearly shows an angle dependence, but none was discussed in this manuscript. Does the Kapitza resistance depend on mismatch angle in additionto grain size?
Reply:This is an interesting question and the answer is definitely positive -- the Kapitza resistance depends on the misorientation angle if we consider individual grain boundaries (GBs). We have in fact carried out a separate study on this question, and find that when the misorientation angle between two grains is relatively large (around 30 degrees), the Kapitza resistance is relatively weakly dependent on the mismatch angle, with the Kapitza conductance after quantum corrections being about 10 W/m^2/K. This is in excellent agreement with the effective Kapitza conductance (9.5 W/m^2/K) obtained in this work. These results will be discussed in a future publication.
Action: We have added the following sentences to the last paragraph ofpage 10 (Just before the Methods section):
“This argument is further supported by the fact that while the Kapitza conductance of individual grain boundaries depends on the angle of misorientation, for angles between about 20 and 40 degrees, this dependence is rather weak. We have carried out a comprehensive study of the Kapitza conductance for grain boundaries of different orientations and the results will be published elsewhere.”
2) CVD grow graphene has a broad distribution of grain sizes. How does this work account for the broad non-uniformity of grain sizes? How about anisotropy in grains that are not nearly circular? If there is anisotropy, asingle or even bimodal Kaptiza resistance cannot be extracted.
Reply: We have considered in this work relatively but not exactly uniformgrain sizes generated by the phase-field crystal method, which has been shown to reproduce realistic grain size distributions in the asymptotic limit in two dimensions [R. Backofen, K. Barmak, K. R. Elder and A. Voigt, Acta Mater. 64, 72 (2014)]. However, we argue that CVD type of non-uniformities and anisotropies could certainly influence theresults quantitatively, but not qualitatively. If the mismatch angledistribution is such that there’s no abundance of small angle GBs, there’s very little change in the Kapitza conductance as explained above in our reply to the Referee’s first question.
We stress that the concept of an effective grain size is very much the same as that of an effective phonon mean free path which, in spite of being a relatively crude estimate, captures the essential physics. Because the in-plane and out-of-plane phonons have drastically distinct transport properties, we expect that the bimodal scaling would survive even if enhanced variations
in the non-uniformity and anisotropy of the GBs were considered.
Action: We have added the following sentences to the last paragraph of page 10 (Just before the Methods section):
“Finally, we point out that our samples were generated by the phase field crystal method, which has been shown to reproduce realistic grain size distributions in the asymptotic limit in two dimensions . Such samples may not correspond to those observed in some experiments , but additional non-uniformity andanisotropy should influence the results only quantitatively, not qualitatively.The concept of an effective grain size is analogous to that of an effective phonon mean free path, which, in spite of being a relatively crude estimate, captures the essential physics. Because the in-plane and out-of-plane phonons have drastically distinct transport properties, we expect that the bimodal scaling would survive even if the influence of additional non-uniformity andanisotropy were taken into account.”
3) Substrate effects are ignored here--most of the time, a CVD grownsample is supported or transferred, and the substrate is known to have adramatic effect on flexural (out of plane) modes. How would your results differ if the simulated samples were on SiO2? If the ZA branch is suppressed by substrate interactions, would the bimodal features nearly disappear?
Reply: We indeed expect that the substrate will have strong effects onthe heat transport
in the systems we have studied. When the ZA branch is suppressed by the interactions with the substrate, the bimodal features may be significantly reduced. To figure out whether it will disappear completely requires detailed study and we leave this interesting question for future work. However, we stress that the suspended case is of critical interest and is the starting point for more elaborated studies.
Action: We have added the following sentences to the last paragraph ofpage 10 (Just before the Methods section):
“We note that we have only considered suspended graphene samples in this work. For supported graphene, heat transport by the out-of-plane phonons will be significantly suppressed. Whether or not the bimodal scaling will survive in the presence of a substrate is an interesting question which requires further study.”
4) corrugation plays a role, and the corrugation here is natural, caused by strain due to mismatch at the GB. In supported samples, the corrugation is also caused by substrate roughness. How would that effect theoutcome?
Reply: This point is related to the previous question. We agree that corrugation caused by a rough substrate can have significant effects on the results and hope to address this question in the future.
5) quantum corrections are performed for the heat capacity but not for the anharmonic rates. How significant is the role of anharmonicity here? Is grain boundary scattering dominant?
Reply: Actually, we have applied quantum corrections to both the heat capacity in the limit where grain boundary scattering dominates and anharmonicity in the limit where phonon-phonon scattering dominates. The difference is that we used a more rigorous mode-to-mode approach to correct the heat capacity in polycrystalline graphene and hence the Kapitza conductance, but used an empirical way to correct the anharmonicity in pristine graphene by scaling the out-of-plane thermal conductivity such that the total thermal conductivity matches a reference value measured experimentally. The reason is that there is so far no rigorous and practical quantum correction method in the diffusive regime where phonon-phonon scattering dominates. In summary, in the limit of polycrystalline graphene with relatively small grains, the grain boundary scattering dominates and one can apply the mode-to-mode method to properly correct the calculated classical Kapitza conductance; in the limit of pristine graphene, phonon-phonon scattering dominates and we can only correct the classical anharmonicity in an empirical way.