Exfoliated monolayer graphene flakes were embedded in a polymer matrix and loaded under axial compression. By monitoring the shifts of the 2D Raman phonons of rectangular flakes of various sizes under load, the critical strain to failure was determined. Prior to loading care was taken for the examined area of the flake to be free of residual stresses. The critical strain values for first failure were found to be independent of flake size at a mean value of -0.60% corresponding to a yield stress up to -6 GPa. By combining Euler mechanics with a Winkler approach, we show that unlike buckling in air, the presence of the polymer constraint results in graphene buckling at a fixed value of strain with an estimated wrinkle wavelength of the order of 1-2 nm. These results were compared with DFT computations performed on analogue coronene/PMMA oligomers and a reasonable agreement was obtained.

] Department of Materials Science University of Patras Rio Patras 26504

] Institute of Chemical Engineering Sciences Foundation of Research and Technology Hellas

] Laboratory of Bio Inspired and Graphene Nanomechanics Department of Civil Environmental and Mechanical Engineering Università di Trento via Mesiano 77 1 38123 Trento Italy [2] Center for Materials and Microsystems Fondazione Bruno Kessler Via Sommarive 18 38123 Povo Italy [3] School of Engineering and Materials Science Queen Mary University of London Mile End Road London E1 4NS UK

Institute of Chemical Engineering Sciences Foundation of Research and Technology Hellas

J Heyrovsky Institute of Physical Chemistry v v i Academy of Sciences of the Czech Republic 182 23 Prague 8 Czech Republic

School of Physics and Astronomy University of Manchester Manchester U K

Zobrazit více v PubMed

Zhao Q. Z., Nardelli M. B. & Bernholc J. Ultimate strength of carbon nanotubes: A theoretical study. Phys Rev B 65, 144105 (2002).

Lee C., Wei X. D., Kysar J. W. & Hone J. Measurement of the elastic properties and intrinsic strength of monolayer graphene. Science 321, 385–388 (2008). PubMed

Frank O. et al. Compression Behavior of Single-Layer Graphenes. Acs Nano 4, 3131–3138 (2010). PubMed

Tsoukleri G. et al. Subjecting a Graphene Monolayer to Tension and Compression. Small 5, 2397–2402 (2009). PubMed

Huang M. Y., Yan H. G., Chen C. Y., Song D. H., Heinz T. F. & Hone J. Phonon softening and crystallographic orientation of strained graphene studied by Raman spectroscopy. P Natl Acad Sci USA 106, 7304–7308 (2009). PubMed PMC

Timoshenko S. Theory of elastic stability, 2d edn. McGraw-Hill. (1961).

Ru C. Q. Column buckling of multiwalled carbon nanotubes with interlayer radial displacements. Phys Rev B 62, 16962–16967 (2000).

Ru C. Q. Axially compressed buckling of a doublewalled carbon nanotube embedded in an elastic medium. J Mech Phys Solids 49, 1265–1279 (2001).

Murmu T. & Pradhan S. C. Vibration analysis of nano-single-layered graphene sheets embedded in elastic medium based on nonlocal elasticity theory. J Appl Phys 105, 064319 (2009).

Pradhan S. C. & Murmu T. Small scale effect on the buckling analysis of single-layered graphene sheet embedded in an elastic medium based on nonlocal plate theory. Physica E 42, 1293–1301 (2010).

Narendar S. & Gopalakrishnan S. Nonlocal continuum mechanics based ultrasonic flexural wave dispersion characteristics of a monolayer graphene embedded in polymer matrix. Compos Part B-Eng 43, 3096–3103 (2012).

Shi J. X., Ni Q. Q., Lei X. W. & Natsuki T. Nonlocal vibration of embedded double-layer graphene nanoribbons in in-phase and anti-phase modes. Physica E 44, 1136–1141 (2012).

Shi J. X., Ni Q. Q., Lei X. W. & Natsuki T. Wave propagation in embedded double-layer graphene nanoribbons as electromechanical oscillators. J Appl Phys 110, 084321 (2011).

Adali S. Variational Principles for Nonlocal Continuum Model of Orthotropic Graphene Sheets Embedded in an Elastic Medium. Acta Math Sci 32, 325–338 (2012).

Kumaki J., Kawauchi T., Okoshi K., Kusanagi H. & Yashima E. Supramolecular helical structure of the stereocomplex composed of complementary isotactic and syndiotactic poly(methyl methacrylate)s as revealed by atomic force microscopy. Angew Chem Int Edit 46, 5348–5351 (2007). PubMed

Novoselov K. S. et al. Electric field effect in atomically thin carbon films. Science 306, 666–669 (2004). PubMed

Gong L., Kinloch I. A., Young R. J., Riaz I., Jalil R. & Novoselov K. S. Interfacial Stress Transfer in a Graphene Monolayer Nanocomposite. Adv Mater 22, 2694–2697 (2010). PubMed

Brangwynne C. P. et al. Microtubules can bear enhanced compressive loads in living cells because of lateral reinforcement. J Cell Biol 173, 733–741 (2006). PubMed PMC

Taj M. & Zhang J. Q. Buckling of embedded microtubules in elastic medium. Appl Math Mech-Engl 32, 293–300 (2011).

Winkler E. Vortriige iiber Eisenbahnbau. (1867).

Castro Neto A. H., Guinea F., Peres N. M. R., Novoselov K. S. & Geim A. K. The electronic properties of graphene. Rev Mod Phys 81, 109–162 (2009).

Aitken Z. H. & Huang R. Effects of mismatch strain and substrate surface corrugation on morphology of supported monolayer graphene. J Appl Phys 107, 123531 (2010).

Tapaszto L., Dumitrica T., Kim S. J., Nemes-Incze P., Hwang C. & Biro L. P. Breakdown of continuum mechanics for nanometre-wavelength rippling of graphene. Nat Phys 8, 739–742 (2012).

Shan W. L., Chen Z., Broedersz C. P., Gumaste A. A., Soboyejo W. O. & Brangwynne C. P. Attenuated short wavelength buckling and force propagation in a biopolymer-reinforced rod. Soft Matter 9, 194–199 (2013).

Xiao J. F. & Chen X. Buckling morphology of an elastic beam between two parallel lateral constraints: implication for a snake crawling between walls. J R Soc Interface 10, 20130399 (2013). PubMed PMC

Zhang K. & Arroyo M. Adhesion and friction control localized folding in supported graphene. J Appl Phys 113, 193501 (2013).

Horinouchi A. & Tanaka K. An effect of stereoregularity on the structure of poly(methyl methacrylate) at air and water interfaces. Rsc Adv 3, 9446–9452 (2013).

Schafer A., Horn H. & Ahlrichs R. Fully Optimized Contracted Gaussian-Basis Sets for Atoms Li to Kr. J Chem Phys 97, 2571–2577 (1992).

Schomaker E. & Challa G. Complexation of Stereoregular Poly(Methyl Methacrylates) .14. The Basic Structure of the Stereocomplex of Isotactic and Syndiotactic Poly(Methyl Methacrylate). Macromolecules 22, 3337–3341 (1989).

Grimme S. Semiempirical GGA-type density functional constructed with a long-range dispersion correction. J Comput Chem 27, 1787–1799 (2006). PubMed

Grimme S. Semiempirical hybrid density functional with perturbative second-order correlation. J Chem Phys 124, 034108 (2006). PubMed

Grimme S. Improved second-order Moller-Plesset perturbation theory by separate scaling of parallel- and antiparallel-spin pair correlation energies. J Chem Phys 118, 9095–9102 (2003).

Distasio R. A. & Head-Gordon M. Optimized spin-component scaled second-order Moller-Plesset perturbation theory for intermolecular interaction energies. Mol Phys 105, 1073–1083 (2007).

Rezac J., Riley K. E. & Hobza P. S66: A Well-balanced Database of Benchmark Interaction Energies Relevant to Biomolecular Structures. J Chem Theory Comput 7, 2427–2438 (2011). PubMed PMC

Weigend F. & Ahlrichs R. Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy. Phys Chem Chem Phys 7, 3297–3305 (2005). PubMed

Boys S. F. & Bernardi F. Calculation of Small Molecular Interactions by Differences of Separate Total Energies - Some Procedures with Reduced Errors. Mol Phys 19, 553–566 (1970).

Morgado C. A., Jurecka P., Svozil D., Hobza P. & Sponer J. Reference MP2/CBS and CCSD(T) quantum-chemical calculations on stacked adenine dimers. Comparison with DFT-D, MP2.5, SCS(MI)-MP2, M06-2X, CBS(SCS-D) and force field descriptions. Phys Chem Chem Phys 12, 3522–3534 (2010). PubMed

Peverati R. & Baldridge K. K. Implementation and Performance of DFT-D with Respect to Basis Set and Functional for Study of Dispersion Interactions in Nanoscale Aromatic Hydrocarbons. J Chem Theory Comput 4, 2030–2048 (2008). PubMed

Cao Y. & Hutchinson J. W. “Wrinkling phenomena in neo-Hookean film substrate bilayers”. Journal of Applied Mechanics 79, 031019 (2012).

Hutchinson J. W. “The role of nonlinear substrate analysis in the wrinkling of thin films”. Phil. Trans. R. Soc. A 371, 20120422 (2013). PubMed

Biot M. A. “Bending of infinite beam on an elastic foundation”, Harvard school, Publications from the graduate school of engineering, Reprint from Journal of Applied Mechanics. (1937).

Puntel E., Deseri L. & Fried E. “Wrinkling of a stretched thin sheet”. J. Elasticity 105, 137–170 (2011).

Mastering the Wrinkling of Self-supported Graphene

Graphene wrinkling induced by monodisperse nanoparticles: facile control and quantification