Letter pubs.acs.org/NanoLett
Engineering Curvature in Graphene Ribbons Using Ultrathin Polymer Films Chunyu Li,† Marisol Koslowski,‡ and Alejandro Strachan*,† †
School of Materials Engineering and Birck Nanotechnology Center, ‡School of Mechanical Engineering, Purdue University, West Lafayette, Indiana 47906, United States S Supporting Information *
ABSTRACT: We propose a method to induce curvature in graphene nanoribbons in a controlled manner using an ultrathin thermoset polymer in a bimaterial strip setup and test it via molecular dynamics (MD) simulations. Continuum mechanics shows that curvature develops to release the residual stress caused by the chemical and thermal shrinkage of the polymer during processing and that this curvature increases with decreasing film thickness; however, significant deformation is only achieved for ultrathin polymer films. Quite surprisingly, explicit MD simulations of the curing and annealing processes show that the predicted trend not just continues down to film thicknesses of 1−2 nm but that the curvature development is enhanced significantly in such ultrathin films due to surface tension effects. This combination of effects leads to very large curvatures of over 0.14 nm−1 that can be tuned via film thickness. This provides a new avenue to engineer curvature and, thus, electromagnetic properties of graphene. KEYWORDS: Graphene, nanoribbon, band structure engineering, curvature, molecular dynamics
S
ince the initial exfoliation studies,1 graphene has attracted much attention due to its outstanding transport properties and strength.2,3 Significant progress has been made in the growth of graphene and other two-dimensional (2D) materials and their integration into electronic, photonics, and optoelectronic devices is advancing at a fast pace.4,5 A perfect graphene sheet is a semimetal with zero band gap and the ability to open a gap and tune electronic properties is highly desirable for a wide range of applications. Consequently, several avenues have been explored to engineer graphene’s electronic properties. For example, nanopatterning this 2D material into ribbons affects its electronic response due to quantum confinement and the effect of surface states.2 Another approach involves the selective adsorption of molecules.6 Absorbed molecules can act as defect centers resulting in the opening of a band gap. A third avenue involves the application of external mechanical, electrical, or magnetic stimuli to control graphene’s response.7−9 Motivated by the possibility of engineering response via strain and curvature, significant efforts have been devoted toward the understanding of how these variables affect the electronic response of graphene ribbons.8,10−16 Several approaches have been proposed to control the global or local curvature of graphene as well as to induce in-plane strain. DeParga et al.17 reported the growth of periodically rippled graphene monolayers on a single crystal substrate, and Bao et al.18 demonstrated controlling of the periodic ripples in suspended graphene sheets by thermomechanical manipulation. Graphene bubbles raised over the substrate have also been found, characterized,19 and controlled by electric fields.20 Mu et al.21 investigated theoretically the voltage dependence of the © XXXX American Chemical Society
nanobubble radius. Chemi- and physisorption of small molecules on graphene can also lead to bending with the resulting curvature depending on coverage and nature of the interactions.22 The interaction of graphene with small liquid droplets can also lead to significant bending due to surface tension effects.23,24 Despite these advances, inducing large, controlled, and permanent curvature in graphene by physical means remains elusive. In this paper, we demonstrate that this can be achieved using thin films of thermoset polymers deposited over graphene ribbons into bimaterial configurations. Ultrahigh curvatures, over ∼0.14 nm−1, are achievable due to a combination of effects, including surface tension of the liquid resin and a significant buildup of residual stress originating from a combination chemical shrinkage during polymer cure and the difference in thermal expansion coefficient between the two materials. The difference in thermomechanical response between polymers and carbon-based materials is key in the design of advanced composites for structural applications in aerospace, energy, automotive, and other applications.25 However, the difference in thermal expansion coefficients poses significant challenges in the processing of composites because the resulting internal stresses degrade performance.26 This effect is compounded when thermoset polymers are used because chemical shrinkage during curing adds to the development of residual stresses.27 In this paper, we make use of the often Received: September 13, 2014 Revised: October 27, 2014
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dx.doi.org/10.1021/nl503527w | Nano Lett. XXXX, XXX, XXX−XXX
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question now is whether such a thin epoxy thermoset film could be cured to achieve high levels of cross-linking and how size effects on the thermomechanical properties of thin polymer films, such as glass transition depression and softening,30,31 would affect the development of curvature. To test the feasibility of the approach and the predictions of the continuum model we performed explicit molecular dynamics (MD) simulations of the complete process. We start by depositing an ultrathin film of a thermoset resin (epoxy monomers mixed with a curing agent) on the surface of a freestanding graphene nanoribbon; we then simulate in situ curing of the polymer and finally anneal the system to room temperature. The MD results not only confirm that a significant curvature can be induced using ultrathin films but show that the continuum model significantly underestimates the curvature for ultrathin films, see symbols in Figure 1. The simulations shows that remarkably large curvatures of over 0.14 nm−1 are attainable for film with thickness ∼2.8 nm. Molecular Dynamics Simulation Details. We performed MD simulations on freestanding graphene nanoribbons of lengths ranging from ∼10 to ∼30 and ∼5 nm in width; the ribbons are oriented with the armchair crystallographic direction along their length. The epoxy resins (a mixture DGEBF and DETDA, see the Supporting Information Material for their molecular structures) are deposited on one side of the graphene ribbons. Each of the two amine groups in DETDA can react with two epoxy molecules and each epoxy can bridge between two amines. Thus, our simulations use a perfect stoichiometry of 2:1 amine to epoxy ratio that in principle could enable 100% conversion; in practice this limit is not reached, especially for thin film configurations. Atomic interactions are described using the DREIDING force field32 with partial atomic charges obtained using the electrochemical potential equalization method.33 All MD simulations are carried out using LAMMPS,34 a massively parallel MD simulator from Sandia National Laboratories and additional simulation details are included in the Supporting Information. The graphene/uncured resin systems are equilibrated using isothermal MD simulations at T = 600 K for 400 ps after which the resin is cured. For the latter step, we use the MD-based Polymerization Simulator (MDPoS) method, described in detail in ref 35, which has been shown to produce realistic molecular structures and to lead to accurate predictions of thermo-mechanical properties. MDPoS mimics the polymerization by the periodic creation of bonds between pairs of reactive atoms using a distance criterion (with a cutoff equal to four times of the equilibrium N−C bond length of 1.41 Å). The new chemical bonds are turned on slowly using a 50 ps long multistep relaxation procedure to avoid large atomic forces. After the new set of bonds is fully relaxed, the system is thermalized for an additional 50 ps and a new round of bond creation is started. This procedure is carried out at 600 K for a total simulation time of 2 ns; in all cases, the thin films achieve conversion degrees of 75% or higher. The cross-linked systems are then annealed from the curing temperature (600 K) to room temperature using a stepwise cooling process at a rate of 10 K/60 ps. No chemical reactions are allowed between the polymer and the inert graphene ribbon and charge transfer between the polymer and graphene is neglected; thus, the interactions between them originate purely from van der Waals forces (London dispersion plus Pauli repulsion). The combination of MDPoS with this interatomic potential has been shown to provide an accurate description of the key
undesired significant internal stress build up and its relaxation to engineer curvature in graphene using a bimaterial setup with ultrathin epoxy films. Development of Curvature in a Bimaterial Strip. The expected curvature development as a function of film thickness can be estimated using Timoshenko’s bimaterial model.28 This model describes the curvature development due to misfit strain using linear elasticity and a beam model for the elastic response of the graphene and polymer films and assuming no interfacial slip. The effect of the free surface energy on the total strain energy is important in nanoscale specimens and it has been incorporated to Timoshenko’s description.29 The curvature in the modified Timoshenko model is29 κ=6
εmEf Estf ts(tf + ts) + G(Ef tf2 + Ests(2t f + ts)) Ef2tf4 + ES2tS4 + 2Ef Estf ts(2tf2 + 3tf ts + 2ts2)
(1)
where εm is the misfit strain, E is the Young’s modulus, G is the free surface energy of the polymer, t represents thickness, and the subscripts f and s indicate the film (polymer) and the substrate (graphene). When the surface energy, G, is set to zero the original curvature derived by Timoshenko is recovered.28 This beam solution corresponds to plane stress conditions; if instead we consider a plate with finite thickness, the biaxial modulus E/1 − ν, with ν being Poisson’s ratio, should be used instead of E in eq 1. Full lines in Figure 1 show the estimated curvature as a function of film thickness for plane stress and plane strain
Figure 1. Curvature versus film thickness. Solid lines are obtained with the modified Timoshenko solution with nominal values of materials properties for plain stress and plain strain conditions; see Section 1 of Supporting Information. Symbols are obtained from MD simulations.
conditions and with all materials properties obtained from MD simulations for resins of diglycidyl ether of bisphenol F (DGEBF) epoxy and curing agent diethylenetoluenediamine (DETDA). All model input parameters are included in Table S1 of the Supporting Information where we also discuss the accuracy of the predicted properties and provide additional details of the continuum model. The predictions use a volumetric chemical shrinkage of εv = −6% (this only accounts for the densification for cure degrees over the gel point as the molten resin can relax internal stresses via flow prior to that) and εv = −10.1% volumetric strain from thermal expansion, see Supporting Information. These values are also obtained MD simulations that show excellent agreement with experiments.27 The continuum model predicts significant curvature but only for very thin resins with a maximum curvature of ∼0.05 nm−1 for a polymer film of thickness 1.24 nm (see Section 1 of the Supporting Information for details of the calculations). The B
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Figure 2. Atomistic snapshots of the graphene/epoxy bimaterials after cross-linking at 600 K for the various ribbon lengths and film thicknesses. The development of curvature for the thin film cases can be clearly seen.
Figure 3. Atomistic snapshots of the graphene/epoxy bimaterials after annealing to T = 300 K for the various ribbon lengths and film thicknesses. The development of curvature for the thin film cases can be clearly seen.
properties of thermoset polymers to predict the development of curvature of interest in this paper. These include the glass transition temperature, stiffness, chemical shrinkage, and thermal expansion coefficient36,37 and captures size effects on the thermomechanical properties of thin films.38 We have recently demonstrated that MD-derived properties can be used to predict the processing-induced development of internal stresses in an epoxy-composite bimaterial setup in excellent agreement with experiments.25 The approach also leads to accurate description of the stiffness and thermal expansion coefficient for graphene; see Supporting Information. Curvature Development during Curing. As discussed above, the volume of the polymer shrinks during processing both due to the chemical reactions during cure as well as due to the thermal contraction during annealing.39 Chemical reactions between the primary amine and the epoxy group of the monomer increase the length or branch polymer chains, reducing the number of chain-ends by two. An epoxy reaction with a secondary amine removes one chain end. Thus, each of these chemical reactions reduces the volume of the polymer and as we will see below leads to residual stresses. Figure 2 show atomistic snapshots of the systems at T = 600 K after conversion. Despite significant thermal fluctuations, we observe quite a large development of curvature for the systems with the thinnest polymer films. As shown in the Supporting Information, the simulations show a monotonic increase in curvature with cure degree but, interestingly, the development
of curvature is rather weak for low conversion degrees and increases significantly for conversion degrees larger than 40− 60%. The reason for this is the complex processes that the resin undergoes during cure. While the rate of chemical volume shrinkage is rather independent of cure degree,40 the initial resin is in a liquid state and the shrinkage can be accommodated, at least partially, by flow. During curing, the viscous liquid transforms into a rubber at the gel point and its ability to transfer loads increases significantly. The gel point corresponds approximately to 60% conversion for the polymers of interest here (in their bulk form) coinciding with the onset of significant curvature development. Also, worthy of note in Figure 2 is the observed interfacial sliding between the resin and the graphene ribbons. This additional mechanism of stress relaxation that does not lead to curvature is explicitly captured in the MD simulations; we find that their effect is not significant. Curvature Development during Annealing. The increase in curvature during the annealing process from the curing temperature to room temperature is very noticeable. Figure 3 shows snapshots of the epoxy/graphene bilayer systems at T = 300 K and Figure 4 shows the curvature as a function of temperature for selected cases. As predicted by the continuum model, our MD simulations show that decreasing film thickness leads to increased curvature. More interestingly, the simulations show that curvatures of over 0.14 nm−1 are achievable in very thin slabs; this is significantly higher than the C
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and the Boeing Co. Computational resources from nanoHUB are gratefully acknowledged. The authors acknowledge useful discussions with R. B. Pipes.
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(1) Novoselov, K. S.; Jiang, D.; Schedin, F.; Booth, T. J.; Khotkevich, V. V.; Morozov, S. V.; Geim, A. K. Two-dimensional atomic crystals. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 10451−10453. (2) Castro Neto, A. H.; Guinea, F.; Peres, N. M. R.; Novoselov, K. S.; Geim, A. K. The electronic properties of graphene. Rev. Mod. Phys. 2009, 81 (1), 109−162. (3) Schwierz, F. Graphene transistors. Nat. Nanotechnol. 2010, 5 (7), 487−496. (4) Westervelt, R. M. Graphene Nanoelectronics. Science 2008, 320 (5874), 324−325. (5) Bonaccorso, F.; Sun, Z.; Hasan, T.; Ferrari, A. C. Graphene photonics and optoelectronics. Nat. Photonics 2010, 4 (9), 611−622. (6) Berashevich, J.; Chakraborty, T. Tunable band gap and magnetic ordering by adsorption of molecules on graphene. Phys. Rev. B 2009, 80 (3), 033404. (7) Pereira, V. M.; Neto, A. H. C. Strain Engineering of Graphene’s Electronic Structure. Phys. Rev. Lett. 2009, 103 (4), 4. (8) Hod, O.; Scuseria, G. E. Electromechanical Properties of Suspended Graphene Nanoribbons. Nano Lett. 2009, 9 (7), 2619− 2622. (9) Terrones, M.; Botello-Mendez, A. R.; Campos-Delgado, J.; Lopez-Urias, F.; Vega-Cantu, Y. I.; Rodriguez-Macias, F. J.; Elias, A. L.; Munoz-Sandoval, E.; Cano-Marquez, A. G.; Charlier, J. C.; Terrones, H. Graphene and graphite nanoribbons: Morphology, properties, synthesis, defects and applications. Nano Today 2010, 5 (4), 351−372. (10) Glukhova, O.; Slepchenkov, M. Influence of the curvature of deformed graphene nanoribbons on their electronic and adsorptive properties: theoretical investigation based on the analysis of the local stress field for an atomic grid. Nanoscale 2012, 4 (11), 3335−3344. (11) Lu, Y.; Guo, J. Band Gap of Strained Graphene Nanoribbons. Nano Res. 2010, 3 (3), 189−199. (12) Rasmussen, J. T.; Gunst, T.; Boggild, P.; Jauho, A. P.; Brandbyge, M. Electronic and transport properties of kinked graphene. Beilstein J. Nanotechnol. 2013, 4, 103−110. (13) Lehmann, T.; Ryndyk, D. A.; Cuniberti, G. Combined effect of strain and defects on the conductance of graphene nanoribbons. Phys. Rev. B 2013, 88 (12), 6. (14) Zhu, W. J.; Low, T.; Perebeinos, V.; Bol, A. A.; Zhu, Y.; Yan, H. G.; Tersoff, J.; Avouris, P. Structure and Electronic Transport in Graphene Wrinkles. Nano Lett. 2012, 12 (7), 3431−3436. (15) Ni, Z. H.; Yu, T.; Lu, Y. H.; Wang, Y. Y.; Feng, Y. P.; Shen, Z. X. Uniaxial Strain on Graphene: Raman Spectroscopy Study and BandGap Opening. ACS Nano 2008, 2, 2301−2305. (16) Atanasov, V.; Saxena, A. Tuning the electronic properties of corrugated graphene: Confinement, curvature, and band-gap opening. Phys. Rev. B 2010, 81 (20), 205409. (17) deParga, A. L. V.; Calleja, F.; Borca, B.; Passeggi, M. C. G.; Hinarejos, J. J.; Guinea, F.; Miranda, R. Periodically Rippled Graphene: Growth and Spatially Resolved Electronic Structure. Phys. Rev. Lett. 2008, 100 (5), 056807. (18) Bao, W. Z.; Miao, F.; Chen, Z.; Zhang, H.; Jang, W. Y.; Dames, C.; Lau, C. N. Controlled ripple texturing of suspended graphene and ultrathin graphite membranes. Nat. Nanotechnol. 2009, 4 (9), 562− 566. (19) Zabel, J.; Nair, R. R.; Ott, A.; Georgiou, T.; Geim, A. K.; Novoselov, K. S.; Casiraghi, C. Raman Spectroscopy of Graphene and Bilayer under Biaxial Strain: Bubbles and Balloons. Nano Lett. 2012, 12, 617−621. (20) Georgiou, T.; Britnell, L.; Blake, P.; Gorbachev, R. V.; Gholinia, A.; Geim, A. K.; Casiraghi, C.; Novoselov, K. S. Graphene bubbles with controllable curvature. Appl. Phys. Lett. 2011, 99 (9), 093103.
Figure 4. Curvature as a function of temperature from our MD simulations. The lower panel shows the significant increase in curvature as film thickness is decreased.
predictions of the continuum model. To uncover the origin of such discrepancy we analyzed the structure of the bimaterial setups before cross-linking. We observe that surface tension in the liquid thin films leads to the development of curvature even before cross-linking; see a snapshot in Figure S3 of the Supporting Information. This is induced by surface tension and the effect has been shown to produce significant changes in the shape of graphene nanostructures.41 This effect increases with decreasing film thickness and explains the discrepancy between the MD predictions and the continuum model (that ignores relaxations in the liquid state) under those conditions. In summary, we proposed and characterized a method to induce controlled curvature on graphene nanoribbons by load transfer from a thin thermoset polymer film in a bimaterial strip configuration. Our simulations, that make no assumptions regarding load transfer, chemical, or thermal volume shrinkage other than those implicit in the force field description, show that ultralarge curvatures are achievable with thin resin films. The freestanding graphene nanoribbons studied in the simulations are an idealized case but the approach is applicable to other geometries. The use of solid polymers opens the possibility of engineering permanent curvature in suspended nanosheets. We envision the possibility of patterning graphene with thin resin lines and dots to create complex curvature landscapes to obtained desired optoelectronic response.42,43
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ASSOCIATED CONTENT
S Supporting Information *
Curvature calculation by continuum mechanics, molecular structure of monomers, and MD simulation details. This material is available free of charge via the Internet at http:// pubs.acs.org.
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REFERENCES
AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors acknowledge support by the U.S. National Science Foundation under contract CMMI-0826356 and ACI-1440727 D
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(21) Mu, W. H.; Zhang, G.; Ou-Yang, Z. C. Radius-voltage relation of graphene bubbles controlled by gate voltage. Appl. Phys. Lett. 2013, 103 (5), 053112. (22) Zhang, Z. Q.; Liu, B.; Hwang, K. C.; Gao, H. J. Surfaceadsorption-induced bending behaviors of graphene nanoribbons. Appl. Phys. Lett. 2011, 98 (12), 121909. (23) Niladri, P.; Wang, B.; Král, P. Nanodroplet activated and guided folding of graphene nanostructures. Nano Lett. 2009, 9 (11), 3766− 3771. (24) Mirsaidov, U.; Mokkapati, V. R. S. S.; Bhattacharya, D.; Andersen, H.; Bosman, M.; Ö zyilmaz, B.; Matsudaira, P. Scrolling graphene into nanofluidic channels. Lab Chip 2013, 13 (15), 2874− 2878. (25) Ci, L.; Suhr, J.; Pushparaj, V.; Zhang, X.; Ajayan, P. M. Continuous Carbon Nanotube Reinforced Composites. Nano Lett. 2008, 8, 2762−2766. (26) Shokrieh, M. Residual Stresses in Composite Materials; Woodhead Publishing Limited: Cambridge, U.K., 2014. (27) Kravchenko, O. G.; Li, C. Y.; Strachan, A.; Kravchenko, S. G.; Pipes, R. B. Prediction of the chemical and thermal shrinkage in a thermoset polymer. Composites A 2014, 66, 35−43. (28) Timoshenko, S. Analysis of Bi-Metal Thermostats. J. Opt. Soc. Am. 1925, 11, 233−255. (29) Ji Zang; Liu, Feng Modified Timoshenko formula for bending of ultrathin strained bilayer films. Appl. Phys. Lett. 2008, 92, 021905. (30) Forrest, J. A.; Dalnoki-Veress, K. The glass transition in thin polymer films. Adv. Colloid Interface Sci. 2001, 94, 167−196. (31) Stafford, C. M.; Vogt, B. D.; Harrison, C.; Julthongpiput, D.; Huang, R. Elastic moduli of ultrathin amorphous polymer films. Macromolecules 2006, 39, 5095−5099. (32) Mayo, S. L.; Olafson, B. D.; Goddard, W. A., III DREIDING: A Generic Force Field for Molecular Simulations. J. Phys. Chem. 1990, 94, 8897−8909. (33) Mortier, W. J.; Genechten, K. V.; Gasteiger, J. Electronegativity Equalization: Application and Parameterization. J. Am. Chem. Soc. 1985, 107, 829−835. (34) LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator), open source code, http://www.cs.sandia.gov/∼sjplimp/ lammps.html (accessed November 9, 2014). (35) Li, C. Y.; Strachan, A. Molecular simulations of crosslinking process of thermosetting polymers. Polymer 2010, 51, 6058−6070. (36) Li, C. Y.; Strachan, A. Molecular dynamics predictions of thermal and mechanical properties of thermoset polymer EPON862/ DETDA. Polymer 2011, 52, 2920−2928. (37) Li, C. Y.; Medvedev, G.; Caruthers, J. M.; Strachan, A. Molecular dynamics simulations and experimental studies on glassy and rubbery thermomechanical properties of thermoset EPON825/ 33DDS. Polymer 2012, 53, 4222−4230. (38) Li, C. Y.; Strachan, A. Effect of Thickness on Free-Standing Thermosetting Polymeric Nanofilms by Molecular Dynamics Simulations. Macromolecules 2011, 44, 9448−9454. (39) Hill, R. R., Jr; Muzumdar, S. V.; Lee, L. J. Analysis of volumetric changes of unsaturated polyester resins during curing. Polym. Eng. Sci. 1995, 35, 852−859. (40) Ramosa, J. A.; Paganib, N.; Riccardib, C. C.; Borrajob, J.; Goyanesc, S. N.; Mondragon, I. Cure kinetics and shrinkage model for epoxy-amine systems. Polymer 2005, 46, 3323−3328. (41) Zhu, S.; Li, T. Hydrogenation-Assisted Graphene Origami and Its Application in Programmable Molecular Mass Uptake, Storage, and Release. ACS Nano 2014, 8, 2864−2872. (42) Schmidt, O. G.; Eberl, K. Nanotechnology: Thin solid films roll up into nanotubes. Nature 2001, 410, 168. (43) Cho, J. H.; Datta, D.; Park, S. Y.; Shenoy, V. B.; Gracias, D. H. Plastic Deformation Drives Wrinkling, Saddling, and Wedging of Annular Bilayer Nanostructures. Nano Lett. 2010, 10, 5098−5102.
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dx.doi.org/10.1021/nl503527w | Nano Lett. XXXX, XXX, XXX−XXX
Engineering Curvature in Graphene Ribbons Using Ultrathin Polymer Films Chunyu Li,† Marisol Koslowski,‡ and Alejandro Strachan*,† †
School of Materials Engineering and Birck Nanotechnology Center, ‡School of Mechanical Engineering, Purdue University, West Lafayette, Indiana 47906, United States S Supporting Information *
ABSTRACT: We propose a method to induce curvature in graphene nanoribbons in a controlled manner using an ultrathin thermoset polymer in a bimaterial strip setup and test it via molecular dynamics (MD) simulations. Continuum mechanics shows that curvature develops to release the residual stress caused by the chemical and thermal shrinkage of the polymer during processing and that this curvature increases with decreasing film thickness; however, significant deformation is only achieved for ultrathin polymer films. Quite surprisingly, explicit MD simulations of the curing and annealing processes show that the predicted trend not just continues down to film thicknesses of 1−2 nm but that the curvature development is enhanced significantly in such ultrathin films due to surface tension effects. This combination of effects leads to very large curvatures of over 0.14 nm−1 that can be tuned via film thickness. This provides a new avenue to engineer curvature and, thus, electromagnetic properties of graphene. KEYWORDS: Graphene, nanoribbon, band structure engineering, curvature, molecular dynamics
S
ince the initial exfoliation studies,1 graphene has attracted much attention due to its outstanding transport properties and strength.2,3 Significant progress has been made in the growth of graphene and other two-dimensional (2D) materials and their integration into electronic, photonics, and optoelectronic devices is advancing at a fast pace.4,5 A perfect graphene sheet is a semimetal with zero band gap and the ability to open a gap and tune electronic properties is highly desirable for a wide range of applications. Consequently, several avenues have been explored to engineer graphene’s electronic properties. For example, nanopatterning this 2D material into ribbons affects its electronic response due to quantum confinement and the effect of surface states.2 Another approach involves the selective adsorption of molecules.6 Absorbed molecules can act as defect centers resulting in the opening of a band gap. A third avenue involves the application of external mechanical, electrical, or magnetic stimuli to control graphene’s response.7−9 Motivated by the possibility of engineering response via strain and curvature, significant efforts have been devoted toward the understanding of how these variables affect the electronic response of graphene ribbons.8,10−16 Several approaches have been proposed to control the global or local curvature of graphene as well as to induce in-plane strain. DeParga et al.17 reported the growth of periodically rippled graphene monolayers on a single crystal substrate, and Bao et al.18 demonstrated controlling of the periodic ripples in suspended graphene sheets by thermomechanical manipulation. Graphene bubbles raised over the substrate have also been found, characterized,19 and controlled by electric fields.20 Mu et al.21 investigated theoretically the voltage dependence of the © XXXX American Chemical Society
nanobubble radius. Chemi- and physisorption of small molecules on graphene can also lead to bending with the resulting curvature depending on coverage and nature of the interactions.22 The interaction of graphene with small liquid droplets can also lead to significant bending due to surface tension effects.23,24 Despite these advances, inducing large, controlled, and permanent curvature in graphene by physical means remains elusive. In this paper, we demonstrate that this can be achieved using thin films of thermoset polymers deposited over graphene ribbons into bimaterial configurations. Ultrahigh curvatures, over ∼0.14 nm−1, are achievable due to a combination of effects, including surface tension of the liquid resin and a significant buildup of residual stress originating from a combination chemical shrinkage during polymer cure and the difference in thermal expansion coefficient between the two materials. The difference in thermomechanical response between polymers and carbon-based materials is key in the design of advanced composites for structural applications in aerospace, energy, automotive, and other applications.25 However, the difference in thermal expansion coefficients poses significant challenges in the processing of composites because the resulting internal stresses degrade performance.26 This effect is compounded when thermoset polymers are used because chemical shrinkage during curing adds to the development of residual stresses.27 In this paper, we make use of the often Received: September 13, 2014 Revised: October 27, 2014
A
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question now is whether such a thin epoxy thermoset film could be cured to achieve high levels of cross-linking and how size effects on the thermomechanical properties of thin polymer films, such as glass transition depression and softening,30,31 would affect the development of curvature. To test the feasibility of the approach and the predictions of the continuum model we performed explicit molecular dynamics (MD) simulations of the complete process. We start by depositing an ultrathin film of a thermoset resin (epoxy monomers mixed with a curing agent) on the surface of a freestanding graphene nanoribbon; we then simulate in situ curing of the polymer and finally anneal the system to room temperature. The MD results not only confirm that a significant curvature can be induced using ultrathin films but show that the continuum model significantly underestimates the curvature for ultrathin films, see symbols in Figure 1. The simulations shows that remarkably large curvatures of over 0.14 nm−1 are attainable for film with thickness ∼2.8 nm. Molecular Dynamics Simulation Details. We performed MD simulations on freestanding graphene nanoribbons of lengths ranging from ∼10 to ∼30 and ∼5 nm in width; the ribbons are oriented with the armchair crystallographic direction along their length. The epoxy resins (a mixture DGEBF and DETDA, see the Supporting Information Material for their molecular structures) are deposited on one side of the graphene ribbons. Each of the two amine groups in DETDA can react with two epoxy molecules and each epoxy can bridge between two amines. Thus, our simulations use a perfect stoichiometry of 2:1 amine to epoxy ratio that in principle could enable 100% conversion; in practice this limit is not reached, especially for thin film configurations. Atomic interactions are described using the DREIDING force field32 with partial atomic charges obtained using the electrochemical potential equalization method.33 All MD simulations are carried out using LAMMPS,34 a massively parallel MD simulator from Sandia National Laboratories and additional simulation details are included in the Supporting Information. The graphene/uncured resin systems are equilibrated using isothermal MD simulations at T = 600 K for 400 ps after which the resin is cured. For the latter step, we use the MD-based Polymerization Simulator (MDPoS) method, described in detail in ref 35, which has been shown to produce realistic molecular structures and to lead to accurate predictions of thermo-mechanical properties. MDPoS mimics the polymerization by the periodic creation of bonds between pairs of reactive atoms using a distance criterion (with a cutoff equal to four times of the equilibrium N−C bond length of 1.41 Å). The new chemical bonds are turned on slowly using a 50 ps long multistep relaxation procedure to avoid large atomic forces. After the new set of bonds is fully relaxed, the system is thermalized for an additional 50 ps and a new round of bond creation is started. This procedure is carried out at 600 K for a total simulation time of 2 ns; in all cases, the thin films achieve conversion degrees of 75% or higher. The cross-linked systems are then annealed from the curing temperature (600 K) to room temperature using a stepwise cooling process at a rate of 10 K/60 ps. No chemical reactions are allowed between the polymer and the inert graphene ribbon and charge transfer between the polymer and graphene is neglected; thus, the interactions between them originate purely from van der Waals forces (London dispersion plus Pauli repulsion). The combination of MDPoS with this interatomic potential has been shown to provide an accurate description of the key
undesired significant internal stress build up and its relaxation to engineer curvature in graphene using a bimaterial setup with ultrathin epoxy films. Development of Curvature in a Bimaterial Strip. The expected curvature development as a function of film thickness can be estimated using Timoshenko’s bimaterial model.28 This model describes the curvature development due to misfit strain using linear elasticity and a beam model for the elastic response of the graphene and polymer films and assuming no interfacial slip. The effect of the free surface energy on the total strain energy is important in nanoscale specimens and it has been incorporated to Timoshenko’s description.29 The curvature in the modified Timoshenko model is29 κ=6
εmEf Estf ts(tf + ts) + G(Ef tf2 + Ests(2t f + ts)) Ef2tf4 + ES2tS4 + 2Ef Estf ts(2tf2 + 3tf ts + 2ts2)
(1)
where εm is the misfit strain, E is the Young’s modulus, G is the free surface energy of the polymer, t represents thickness, and the subscripts f and s indicate the film (polymer) and the substrate (graphene). When the surface energy, G, is set to zero the original curvature derived by Timoshenko is recovered.28 This beam solution corresponds to plane stress conditions; if instead we consider a plate with finite thickness, the biaxial modulus E/1 − ν, with ν being Poisson’s ratio, should be used instead of E in eq 1. Full lines in Figure 1 show the estimated curvature as a function of film thickness for plane stress and plane strain
Figure 1. Curvature versus film thickness. Solid lines are obtained with the modified Timoshenko solution with nominal values of materials properties for plain stress and plain strain conditions; see Section 1 of Supporting Information. Symbols are obtained from MD simulations.
conditions and with all materials properties obtained from MD simulations for resins of diglycidyl ether of bisphenol F (DGEBF) epoxy and curing agent diethylenetoluenediamine (DETDA). All model input parameters are included in Table S1 of the Supporting Information where we also discuss the accuracy of the predicted properties and provide additional details of the continuum model. The predictions use a volumetric chemical shrinkage of εv = −6% (this only accounts for the densification for cure degrees over the gel point as the molten resin can relax internal stresses via flow prior to that) and εv = −10.1% volumetric strain from thermal expansion, see Supporting Information. These values are also obtained MD simulations that show excellent agreement with experiments.27 The continuum model predicts significant curvature but only for very thin resins with a maximum curvature of ∼0.05 nm−1 for a polymer film of thickness 1.24 nm (see Section 1 of the Supporting Information for details of the calculations). The B
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Figure 2. Atomistic snapshots of the graphene/epoxy bimaterials after cross-linking at 600 K for the various ribbon lengths and film thicknesses. The development of curvature for the thin film cases can be clearly seen.
Figure 3. Atomistic snapshots of the graphene/epoxy bimaterials after annealing to T = 300 K for the various ribbon lengths and film thicknesses. The development of curvature for the thin film cases can be clearly seen.
properties of thermoset polymers to predict the development of curvature of interest in this paper. These include the glass transition temperature, stiffness, chemical shrinkage, and thermal expansion coefficient36,37 and captures size effects on the thermomechanical properties of thin films.38 We have recently demonstrated that MD-derived properties can be used to predict the processing-induced development of internal stresses in an epoxy-composite bimaterial setup in excellent agreement with experiments.25 The approach also leads to accurate description of the stiffness and thermal expansion coefficient for graphene; see Supporting Information. Curvature Development during Curing. As discussed above, the volume of the polymer shrinks during processing both due to the chemical reactions during cure as well as due to the thermal contraction during annealing.39 Chemical reactions between the primary amine and the epoxy group of the monomer increase the length or branch polymer chains, reducing the number of chain-ends by two. An epoxy reaction with a secondary amine removes one chain end. Thus, each of these chemical reactions reduces the volume of the polymer and as we will see below leads to residual stresses. Figure 2 show atomistic snapshots of the systems at T = 600 K after conversion. Despite significant thermal fluctuations, we observe quite a large development of curvature for the systems with the thinnest polymer films. As shown in the Supporting Information, the simulations show a monotonic increase in curvature with cure degree but, interestingly, the development
of curvature is rather weak for low conversion degrees and increases significantly for conversion degrees larger than 40− 60%. The reason for this is the complex processes that the resin undergoes during cure. While the rate of chemical volume shrinkage is rather independent of cure degree,40 the initial resin is in a liquid state and the shrinkage can be accommodated, at least partially, by flow. During curing, the viscous liquid transforms into a rubber at the gel point and its ability to transfer loads increases significantly. The gel point corresponds approximately to 60% conversion for the polymers of interest here (in their bulk form) coinciding with the onset of significant curvature development. Also, worthy of note in Figure 2 is the observed interfacial sliding between the resin and the graphene ribbons. This additional mechanism of stress relaxation that does not lead to curvature is explicitly captured in the MD simulations; we find that their effect is not significant. Curvature Development during Annealing. The increase in curvature during the annealing process from the curing temperature to room temperature is very noticeable. Figure 3 shows snapshots of the epoxy/graphene bilayer systems at T = 300 K and Figure 4 shows the curvature as a function of temperature for selected cases. As predicted by the continuum model, our MD simulations show that decreasing film thickness leads to increased curvature. More interestingly, the simulations show that curvatures of over 0.14 nm−1 are achievable in very thin slabs; this is significantly higher than the C
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and the Boeing Co. Computational resources from nanoHUB are gratefully acknowledged. The authors acknowledge useful discussions with R. B. Pipes.
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(1) Novoselov, K. S.; Jiang, D.; Schedin, F.; Booth, T. J.; Khotkevich, V. V.; Morozov, S. V.; Geim, A. K. Two-dimensional atomic crystals. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 10451−10453. (2) Castro Neto, A. H.; Guinea, F.; Peres, N. M. R.; Novoselov, K. S.; Geim, A. K. The electronic properties of graphene. Rev. Mod. Phys. 2009, 81 (1), 109−162. (3) Schwierz, F. Graphene transistors. Nat. Nanotechnol. 2010, 5 (7), 487−496. (4) Westervelt, R. M. Graphene Nanoelectronics. Science 2008, 320 (5874), 324−325. (5) Bonaccorso, F.; Sun, Z.; Hasan, T.; Ferrari, A. C. Graphene photonics and optoelectronics. Nat. Photonics 2010, 4 (9), 611−622. (6) Berashevich, J.; Chakraborty, T. Tunable band gap and magnetic ordering by adsorption of molecules on graphene. Phys. Rev. B 2009, 80 (3), 033404. (7) Pereira, V. M.; Neto, A. H. C. Strain Engineering of Graphene’s Electronic Structure. Phys. Rev. Lett. 2009, 103 (4), 4. (8) Hod, O.; Scuseria, G. E. Electromechanical Properties of Suspended Graphene Nanoribbons. Nano Lett. 2009, 9 (7), 2619− 2622. (9) Terrones, M.; Botello-Mendez, A. R.; Campos-Delgado, J.; Lopez-Urias, F.; Vega-Cantu, Y. I.; Rodriguez-Macias, F. J.; Elias, A. L.; Munoz-Sandoval, E.; Cano-Marquez, A. G.; Charlier, J. C.; Terrones, H. Graphene and graphite nanoribbons: Morphology, properties, synthesis, defects and applications. Nano Today 2010, 5 (4), 351−372. (10) Glukhova, O.; Slepchenkov, M. Influence of the curvature of deformed graphene nanoribbons on their electronic and adsorptive properties: theoretical investigation based on the analysis of the local stress field for an atomic grid. Nanoscale 2012, 4 (11), 3335−3344. (11) Lu, Y.; Guo, J. Band Gap of Strained Graphene Nanoribbons. Nano Res. 2010, 3 (3), 189−199. (12) Rasmussen, J. T.; Gunst, T.; Boggild, P.; Jauho, A. P.; Brandbyge, M. Electronic and transport properties of kinked graphene. Beilstein J. Nanotechnol. 2013, 4, 103−110. (13) Lehmann, T.; Ryndyk, D. A.; Cuniberti, G. Combined effect of strain and defects on the conductance of graphene nanoribbons. Phys. Rev. B 2013, 88 (12), 6. (14) Zhu, W. J.; Low, T.; Perebeinos, V.; Bol, A. A.; Zhu, Y.; Yan, H. G.; Tersoff, J.; Avouris, P. Structure and Electronic Transport in Graphene Wrinkles. Nano Lett. 2012, 12 (7), 3431−3436. (15) Ni, Z. H.; Yu, T.; Lu, Y. H.; Wang, Y. Y.; Feng, Y. P.; Shen, Z. X. Uniaxial Strain on Graphene: Raman Spectroscopy Study and BandGap Opening. ACS Nano 2008, 2, 2301−2305. (16) Atanasov, V.; Saxena, A. Tuning the electronic properties of corrugated graphene: Confinement, curvature, and band-gap opening. Phys. Rev. B 2010, 81 (20), 205409. (17) deParga, A. L. V.; Calleja, F.; Borca, B.; Passeggi, M. C. G.; Hinarejos, J. J.; Guinea, F.; Miranda, R. Periodically Rippled Graphene: Growth and Spatially Resolved Electronic Structure. Phys. Rev. Lett. 2008, 100 (5), 056807. (18) Bao, W. Z.; Miao, F.; Chen, Z.; Zhang, H.; Jang, W. Y.; Dames, C.; Lau, C. N. Controlled ripple texturing of suspended graphene and ultrathin graphite membranes. Nat. Nanotechnol. 2009, 4 (9), 562− 566. (19) Zabel, J.; Nair, R. R.; Ott, A.; Georgiou, T.; Geim, A. K.; Novoselov, K. S.; Casiraghi, C. Raman Spectroscopy of Graphene and Bilayer under Biaxial Strain: Bubbles and Balloons. Nano Lett. 2012, 12, 617−621. (20) Georgiou, T.; Britnell, L.; Blake, P.; Gorbachev, R. V.; Gholinia, A.; Geim, A. K.; Casiraghi, C.; Novoselov, K. S. Graphene bubbles with controllable curvature. Appl. Phys. Lett. 2011, 99 (9), 093103.
Figure 4. Curvature as a function of temperature from our MD simulations. The lower panel shows the significant increase in curvature as film thickness is decreased.
predictions of the continuum model. To uncover the origin of such discrepancy we analyzed the structure of the bimaterial setups before cross-linking. We observe that surface tension in the liquid thin films leads to the development of curvature even before cross-linking; see a snapshot in Figure S3 of the Supporting Information. This is induced by surface tension and the effect has been shown to produce significant changes in the shape of graphene nanostructures.41 This effect increases with decreasing film thickness and explains the discrepancy between the MD predictions and the continuum model (that ignores relaxations in the liquid state) under those conditions. In summary, we proposed and characterized a method to induce controlled curvature on graphene nanoribbons by load transfer from a thin thermoset polymer film in a bimaterial strip configuration. Our simulations, that make no assumptions regarding load transfer, chemical, or thermal volume shrinkage other than those implicit in the force field description, show that ultralarge curvatures are achievable with thin resin films. The freestanding graphene nanoribbons studied in the simulations are an idealized case but the approach is applicable to other geometries. The use of solid polymers opens the possibility of engineering permanent curvature in suspended nanosheets. We envision the possibility of patterning graphene with thin resin lines and dots to create complex curvature landscapes to obtained desired optoelectronic response.42,43
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ASSOCIATED CONTENT
S Supporting Information *
Curvature calculation by continuum mechanics, molecular structure of monomers, and MD simulation details. This material is available free of charge via the Internet at http:// pubs.acs.org.
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REFERENCES
AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors acknowledge support by the U.S. National Science Foundation under contract CMMI-0826356 and ACI-1440727 D
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