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Journal Club for February 2017: Nanoscale buckling in 2D materials

yongni's picture

    Graphene as a typical two dimensional (2D) crystal membrane attracts tremendous interest. Geometrical distortion such as nanoscale buckling morphology is widely observed in these 2D materials and is crucial to modulating the electronic properties [1]. The calculated bending stiffness of graphene is found to be the same order as that of the lipid bilayer [2], therefore buckling instability may easily occur in graphene leading to non-planar configurations. Different from the conventional buckling transition, the buckling instabilities in 2D materials show different physical origins, and its buckling pattern could be down to nanoscale. Proper mechanics models of nanoscale buckles for their formation and controllability need to be developed for various engineering applications. A few examples of nanoscale buckling in 2D materials are introduced to hopefully offer a few commonplace remarks.

Buckling of a free-standing 2D material

     The results in Figure 1 show that due to the super flexibility of the free-standing 2D material the nanoscale buckling instability  can be easily triggered by various energy dissipations, such as thermal fluctuation [3,4], relaxations of edge stress [5,6], coherent interface stress [7] and stress field created by topological defects [8-10]. The induced out-of-plane distortion can demonstrate remarkable changes compared to the flat counterparts, such as reductions of dislocation formation energy [7], and Griffith strength [11].


Figure 1  Nanoscale buckling structures in a free-standing layer caused by (a) thermal fluctuation [3], (b) edge stress [5], (c) coherent interface relaxation [7],(d) topological defects [10] 

Buckling of a monolayer on substrates

     When graphene-like 2D materials are bonded to substrates, the nanoscale rippling may still be significant although the contribution by thermal fluctuation is neglectable. Figure 2(a)-(b) demonstrates that if the misfit in lattice parameters and/or thermal expansion coefficient between the 2D material and the substrate induces global compression, the monolayer tends to develop wrinkles or fold localization [12-14]. Interestingly, Figure 2(b) also shows that if the monolayer is under global tension due to the mismatch, emergence of interfacial misfit dislocations described by Frenkel–Kontorova model [15] effectively accommodates the mismatch and localized buckles can arise near the dislocation core where the monolayer is under localized compression [16]. This is the case when Frenkel and Kontorova meet Von Karman, the in-plane deformation due to interfacial misfit dislocations are strongly coupled to out-of-plane deformation. The analytical treatment for the localized buckle is still a challenge. In addition, the results obtained by first principle investigation in Figure 3 show that the local chemical bonding environment may have significant influence on the nanoscale buckling [17,18].

 Figure 2 Sketch of nanoscale buckling structures in a monolayer on substrates (a) ,(b)wrinkling to folding transition under global compression [12,13], (b) formation of ordered rippling pattern under localized compression near the dislocation core [16]


Figure 3  Formation of nanoscale rippled graphene phase on substrates by first principle investigation, (a)bonding energy between the graphene edge and the Ni substrate, (b) total energy change [17], (c),(d) the bonding structures in the rippled graphene on Ir substrate [18]

Buckling of twisted bilayer graphene

In bilayer graphene, twist modifies the electronic properties. The twist stacking usually leads to a characteristic interfacial dislocation pattern, known as a Moiré pattern. Emergence of significant buckling in the bilayer graphene governs further relaxation of the Moiré pattern [19-24]. The results in Figure 4 show that relaxed Moiré patterns with out-of-plane displacement and twisted dislocation structures can form [21,22].


Figure 4  Nanoscale buckling arose in twisted bilayer graphene, (a) simulated result by a generalized Peierls−Nabarro model [22], (b) simulated result by the molecular dynamics code LAMMPS [23]

 The interfacial dislocation with  out-of-plane deformation is a so called strain soliton [25]. The results in Figure 5 show that manipulating and visualizing the motion of nanoscale buckled structures have attracted increasing interest since they have substantial effects on the electronic and mechanical properties of such 2D materials [25,26].

Figure 5 Motion and visualization of nanoscale buckled structures (a) movement of the nanoscale wrinkle by thermophoresis [26] (b) visualization of domain-wall soliton in exfoliated bilayer graphene [25]
In most 2D materials, out-of-plane distortion as an efficient way to relax compressive stress. It usually competes with other stress relieving modes. The different coupling mechanisms may exhibit different physical origins of the observed nanoscale bucklings. Further theoretical investigations into this fascinating system are to be exploited.

1.       Neto A H C, Guinea F, Peres N M R, et al. The electronic properties of graphene[J]. Reviews of modern physics, 2009, 81(1): 109.

 2.       Wei Y, Wang B, Wu J, et al. Bending rigidity and Gaussian bending stiffness of single-layered graphene[J]. Nano letters, 2012, 13(1): 26-30. 

3.       Fasolino A, Los J H, Katsnelson M I. Intrinsic ripples in graphene[J]. Nature materials, 2007, 6(11): 858-861. 

4.       Gao W, Huang R. Thermomechanics of monolayer graphene: Rippling, thermal expansion and elasticity[J]. Journal of the Mechanics and Physics of Solids, 2014, 66: 42-58.  

5.       Shenoy V B, Reddy C D, Ramasubramaniam A, et al. Edge-stress-induced warping of graphene sheets and nanoribbons[J]. Physical review letters, 2008, 101(24): 245501. 

6.       Liu X Y, Wang F C, Wu H A. Anisotropic growth of buckling-driven wrinkles in graphene monolayer[J]. Nanotechnology, 2015, 26(6): 065701. 

7.       Nandwana D, Ertekin E. Ripples, strain, and misfit dislocations: structure of graphene–boron nitride superlattice interfaces[J]. Nano letters, 2015, 15(3): 1468-1475. 

8.       Seung H S, Nelson D R. Defects in flexible membranes with crystalline order[J]. Physical Review A, 1988, 38(2): 1005. 

9.       Lusk M T, Carr L D. Nanoengineering defect structures on graphene[J]. Physical review letters, 2008, 100(17): 175503. 

10.    Zhang T, Li X, Gao H. Defects controlled wrinkling and topological design in graphene[J]. Journal of the Mechanics and Physics of Solids, 2014, 67: 2-13. 

11.    Song Z, Ni Y, Xu Z. Geometrical distortion leads to Griffith strength reduction in graphene membranes[J]. Extreme Mechanics Letters, 2017. 

12.    Aitken Z H, Huang R. Effects of mismatch strain and substrate surface corrugation on morphology of supported monolayer graphene[J]. Journal of Applied Physics, 2010, 107(12): 123531. 

13.    Zhang K, Arroyo M. Adhesion and friction control localized folding in supported graphene[J]. Journal of Applied Physics, 2013, 113(19): 193501. 

14.    Zhang K, Arroyo M. Understanding and strain-engineering wrinkle networks in supported graphene through simulations[J]. Journal of the Mechanics and Physics of Solids, 2014, 72: 61-74. 

15.    Braun O M, Kivshar Y. The Frenkel-Kontorova model: concepts, methods, and applications[M]. Springer Science & Business Media, 2013. 

16.    Lu J, Zhang K, Liu X F, et al. Order–disorder transition in a two-dimensional boron–carbon–nitride alloy[J]. Nature communications, 2013, 4. 

17.    Yuan Q, Hu H, Gao J, et al. Upright standing graphene formation on substrates[J]. Journal of the American Chemical Society, 2011, 133(40): 16072-16079. 

18.    Imam M, Stojić N, Binggeli N. First-Principles Investigation of a Rippled Graphene Phase on Ir (001): The Close Link between Periodicity, Stability, and Binding[J]. The Journal of Physical Chemistry C, 2014, 118(18): 9514-9523. 

19.    Alden J S, Tsen A W, Huang P Y, et al. Strain solitons and topological defects in bilayer graphene[J]. Proceedings of the National Academy of Sciences, 2013, 110(28): 11256-11260. 

20.    Butz B, Dolle C, Niekiel F, et al. Dislocations in bilayer graphene[J]. Nature, 2014, 505(7484): 533-537.

21.    Dai S, Xiang Y, Srolovitz D J. Twisted bilayer graphene: Moiré with a twist[J]. Nano letters, 2016, 16(9): 5923-5927.

22.    van Wijk M M, Schuring A, Katsnelson M I, et al. Relaxation of moiré patterns for slightly misaligned identical lattices: graphene on graphite[J]. 2D Materials, 2015, 2(3): 034010.

23.    Zhou S, Han J, Dai S, et al. van der Waals bilayer energetics: Generalized stacking-fault energy of graphene, boron nitride, and graphene/boron nitride bilayers[J]. Physical Review B, 2015, 92(15): 155438.

24.    Kumar H, Er D, Dong L, et al. Elastic Deformations in 2D van der waals Heterostructures and their Impact on Optoelectronic Properties: Predictions from a Multiscale Computational Approach[J]. Scientific reports, 2015, 5.

25.    Jiang L, Shi Z, Zeng B, et al. Soliton-dependent plasmon reflection at bilayer graphene domain walls[J]. Nature materials, 2016.

26.    Guo Y, Guo W. Soliton-like thermophoresis of graphene wrinkles[J]. Nanoscale, 2013, 5(1): 318-323.





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Cai Shengqiang's picture

Dear Yong, 

Many thanks for such an insightful review. 

After reading your review, I feel that the new opportunities in the wrinkling/bulking of 2D materials may be associated with the new microscopic mechanisms of generating instabilities/deformation, which may be unimportant in conventional materials. 

One question for you, continuum treatment of 2D materials is really a big assumption. It is nice to see many phenomenon in 2D materials can be well explained by continuum mechanics, more specifically, elasticity theory. Do you know any examples showing that the mechanical behaviors of 2D materials cannot be explained by continuum mechanics? 

Again, really enjoy reading the thoughtful and concise review. 


yongni's picture

Dear  Shengqiang

      Thank you for the encouragement. I agree that instability and deformation of 2D materials especially the interactions between 2D materials and the underlying substrates will attract more research interest for future engineering applications. I just noted a very comprehensive review about mechanics of 2D materials.


D. Akinwande, C.J. Brennan, J.S. Bunch, P. Egberts, J.R. Felts, H. Gao, R. Huang, J.-S. Kim, T. Li, Y. Li, K.M. Liechti, N. Lu, H.S. Park, E.J. Reed, P. Wang, B.I. Yakobson, T. Zhang, Y.-W. Zhang, Y. Zhou, Y. Zhu, A Review on Mechanics and Mechanical Properties of 2D Materials - Graphene and Beyond. Extreme Mechanics Letters 13, 42-72 (2017).

I really hope that continuum mechanics can also be applied in the above field. However, a successful continuum model should take into account the emerging physical effects due to discrete lattice effect, nonlinear effect and possible multifield coupling effect. These effects are notable in 2D materials wherein several energetic scales are comparable. 

I just noted one reference that argues the breakdown of continuum mechanics for nanometer-wavelength rippling of graphene.

Tapasztó L, Dumitrică T, Kim S J, et al. Breakdown of continuum mechanics for nanometre-wavelength rippling of graphene[J]. Nature physics, 2012, 8(10): 739-742.


Kejie Zhao's picture

Dear Yong,

Thank you for the very nice review. I am a beginner and primarily interested in the defects and their effect on the physicochemistry of 2D materials. We did some peliminary studies on the interactions of grain boundaries of graphene with corrosive species. I am not aware of many studies on the influence of atomic defects on the mechanical, chemical, interfacial, etc stabilities/functions of 2D materials, could you please provide some insight and guidance on this field?  Thank you.




yongni's picture

Dear Kejie
     What you mentioned about defects in 2D materials is a relatively wide range of issue. Although i am very interested in the mechanics of 2D materials, i as a beginner did little research in this field so far. The following listed reference can definitely provide more insight than me.
D. Akinwande, C.J. Brennan, J.S. Bunch, P. Egberts, J.R. Felts, H. Gao, R. Huang, J.-S. Kim, T. Li, Y. Li, K.M. Liechti, N. Lu, H.S. Park, E.J. Reed, P. Wang, B.I. Yakobson, T. Zhang, Y.-W. Zhang, Y. Zhou, Y. Zhu, A Review on Mechanics and Mechanical Properties of 2D Materials - Graphene and Beyond. Extreme Mechanics Letters 13, 42-72 (2017).

In addition, i am trying to figure out what are important problems in mechanics of 2D materials and what kind of tools we can use or need to develop. It would be my pleasure to have a cup of coffee with you for the discussion and read your newest paper in future. Thanks for your comment.


Kejie Zhao's picture

Dear Yong,  Thanks for recommending the excellent review paper. I will look forward to our meeting next time!  -Kejie

exw569's picture

Thank you for the article. Rearlly very instructive.

I recently started investigating buckling of 2D nanomaterials in flowing liquids. It is a quite complex phenomenon. I have several postdoc positions on the topic coming up, the first one will be open soon


If one of your students is looking for a position, and is interested in the coupling betweeen fluid mechanics and solid mechanics in the deformation of 2D materials, I would appreciate if you could point him/her to the job ad above.

Thank you!

Lorenzo Botto



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