A Buckling Problem for Graphene Sheet J. Galagher 1, Y. Milman 2, S. Ryan 3, D. Golovaty 3, P. Wilber 3, and A. Buldum 4 1 Department of Phyic, Rocheter Intitute of Technology, Rocheter, NY 14623, USA 2 Department of Mathematic, The City College of New York, New York, NY 131, USA 3 Department of Theoretical and Applied Mathematic, The Univerity of Akron, Akron, OH 44325, USA 4 Department of Phyic, The Univerity of Akron, Akron, OH 44325, USA jrg235@rit.edu, milmann@netzero.net, dr17@uakron.edu, dmitry@math.uakron.edu, pwilber@math.uakron.edu, buldum@nebula.phyic.uakron.edu ABSTRACT: We develop a continuum model that decribe the elatic bending of a graphene heet interacting with a rigid ubtrate by van der Waal force. Uing thi model, we tudy a buckling problem for a graphene heet perpendicular to a ubtrate. After identifying a trivial branch, we combine analyi and computation to determine the tability and bifurcation of olution along thi branch. Alo preented are the reult of atomitic imulation. The imulation agree qualitatively with the prediction of our continuum model but alo ugget the importance, for ome problem, of developing a continuum decription of the van der Waal interaction that incorporate information on atomic poition. Keyword: graphene, elatica model, Lennard-Jone potential, tability analyi 1 Introduction A graphene heet i a ingle-atom-thick layer of carbon atom in which each atom i bonded to it three nearet neighbor to form a hexagonal lattice. Becaue tacked graphene heet are the building block of graphite, a naturally occurring form of carbon, there ha been a long-tanding interet in their mechanical propertie. Thi interet ha intenified during the lat decade with the exploion of reearch on carbon nanotube, which have the tructure of a graphene heet rolled into a cylinder. Numerou tudie have demontrated that carbon nanotube have remarkable mechanical propertie (ee [5] for a urvey), and it i expected that iolated graphene heet will hare many of thee propertie. Currently, mechanical and chemical method are being developed for iolating individual graphene layer [1] [4]. The general motivation for thi work i to develop accurate continuum model of graphene. More pecifically, the problem we formulate i driven by recent interet in the fabrication of device at the nanocale [6]. Becaue of their mechanical and electrical propertie, graphene heet may be ueful building block in a variety of potential nanocale device [7] [9]. However, fabricating uch device may entail the ucceful mechanical manipulation of individual graphene heet. One could view our problem a decribing part of a nanocale fabrication proce in which a graphene heet i poitioned perpendicularly againt a rigid urface. (Thi poitioning could be accomplihed by the probe tip of an atomic force microcope [1].) Important for uccefully building the device would be to know how hard the heet can be puhed before buckling into a hape that i no longer perpendicular to the rigid urface. In thi paper, we develop a continuum model of a graphene heet interacting with a rigid ubtrate. The graphene heet i modeled a an elatica. We ue a variational approach, in which the energy ha two part, the bending energy of the heet and the
energy from the interaction of the heet with the ubtrate by van der Waal force. In our continuum model, the van der Waal potential i contructed in uch a way that it depend neither on the poition of the atom on the heet nor on the poition of the atom on the ubtrate. Uing the model, we tudy the buckling of a graphene heet perpendicular to a rigid ubtrate. The bottom edge of the heet approache the ubtrate a the top edge of the heet i loaded by a precribed force. After identifying a trivial branch of olution, we combine analyi and computation to tudy the tability and bifurcation of olution along thi branch. Alo included are the reult of atomitic imulation. Thee imulation agree qualitatively with the prediction of our model. An intereting feature of the atomitic reult i a dicrete jumping, a oppoed to a continuou lipping, of the bottom edge of the buckled heet acro the ubtrate. Specifically, a the load i increaed beyond the buckling load, the heet bottom edge jump between particular atomic location on the ubtrate. Thi phenomena i not predicted by our model, which ugget that a continuum decription of van der Waal interaction hould be developed that incorporate information on atomic poition. 2 Formulation of Model We model a ubtrate a a rigid infinite planar graphene heet. We aume that interacting with thi ubtrate i a flexible graphene heet of length L and infinite width. We conider only deformation of the heet for which it top and bottom edge remain traight and parallel both to the ubtrate and to each other and for which the deformation i the ame in any plane perpendicular to the top and bottom edge. The deformation of a typical cro-ection i depicted in Figure 1. A vertical contact force of linear denity λ i applied to the top edge, while the bottom edge remain free. For a typical material point [, L], we let θ() denote the angle that the tangent to the cro-ection make with the horizontal. We denote the ditance between the bottom edge of the graphene heet (which correpond to = ) and the ubtrate by. λ θ() Figure 1: Graphene heet and ubtrate. We approximate the van der Waal energy of interaction between the ubtrate and the heet uing the Lennard-Jone potential that depend on the ditance between the point on two continuum urface. (We ignore the elf-interaction within each lattice.) Once integrated over the entire ubtrate, the interaction energy at each point of the heet depend only on the ditance between that point and the ubtrate. The graph of the van der Waal energy e per unit length along the cro ection i hown in Figure 3(a). Note that thi profile cloely reemble that of the tandard 6-12 potential. See [11] for detail on the derivation of e. We note that our approximation to the van der Waal interaction i accurate away from the ubtrate, but fail at ditance near to the equilibrium eparation d in Figure 3(b),
where the effect of individual atomic poition begin to be felt. We dicu the conequence of the continuum approximation in the ubequent ection. The energy of the graphene heet conit of the bending energy [12] (a in the elatica model [13]) upplemented with the potential energy of interaction with the ubtrate ( ( β )) E[θ, ] = 2 (θ ) 2 + e + in θ dσ d, (1) where β i the bending contant and e i the energy denity of the van der Waal interaction. The ditance between the point correponding to the arclength and the ubtrate i + in θ dσ. Note that the hape of the heet i completely determined once and θ are known. The graphene heet i vertical if θ π/2. The energy of the perturbed configuration ( + δ, π/2 + δθ) i given by ( ( β )) E[π/2+δθ, +δ ] = 2 (δθ ) 2 + e + δ + co(δθ)dσ d. (2) The potential energy due to an external force of the linear denity λ applied at the top of the heet i ( ) W [δθ, δ ] = λ δ + co(δθ)dσ L. (3) Then the total potential energy E + W of the perturbed configuration i ( ( β )) F [δθ, δ ] := 2 (δθ ) 2 + e + δ + co(δθ)dσ d λ(δ + co(δθ)dσ L). (4) By linearizing F near δ =, δθ, we obtain a force balance condition λ = e( + L) e( ). (5) (Note that θ = π/2 already atifie the appropriate Euler-Lagrange equation.) The tability of the vertical olution can be deduced from the econd variation of the potential energy. Collecting the econd-order term in the expanion of F with repect to δ and δθ and integrating by part in (2), we obtain G[δθ, δ ] := 1 2 3 Stability Analyi ( β(δθ ) 2 + [e( + ) e( )]δθ 2) d (6) + δy2 2 [e ( + L) e ( )]. A vertical olution exit exactly when the contact force λ applied at the top of the heet and the poition of the bottom of the heet atify (5). Figure 2 how the curve of point atifying (5) in the λ -plane for a typical value of L. Thi curve repreent the vertical olution, which correpond to our trivial branch, whoe bifurcation and tability we analyze next. We define a olution on the vertical branch to be table if the econd variation (6) i
y c (L) d 2 y d (L) d d 1 λ Figure 2: Branch of vertical olution poitive emi-definite, i.e., i non-negative for every combination of δθ and δ. e e L d 1 y d (L) d d y c (L) d 2 (a) (b) Figure 3: (a) Graph of energy e per unit length. (b) Graph of e Firt we claim that, given L, there i a y c (L) uch that any olution on the vertical branch with > y c (L) i untable. Thi i eaily verified by examining Figure 3(b), from which one ee that if > y c (L), then e ( + L) e ( ) <. Hence the econd variation G can be made negative by chooing δθ, which make the integral term in (6) vanih, and by chooing any δ. Note from Figure 3(b) that y c (L) > d. Alo, y c (L) d 2, the point at which e attain it maximum, a L, and y c (L) d a L. Next we claim that, given L, there i a y d (L) uch that any olution on the vertical branch with < y d (L) i untable. We define g( ) = [e( + ) e( )] d. Uing Figure 3(a), one check that g( ) < for < d 1, that g( ) > for > d, and that g ( ) > for d 1 d. It follow that there exit a unique value y d (L) with d 1 < y d (L) < d uch that g( ) < for < y d (L) and g( ) > for > y d (L). For < y d (L), we can chooe δ = and δθ equal to any non-zero contant function to make (6) negative, o that the olution correponding to thi i untable. Alo, traightforward argument how that y d (L) d a L and that y d (L) d 1 a L. The argument above how that any table olution on the vertical branch can occur only for between y d (L) and y c (L). Conider between d and y c (L). From Figure 3(b) we ee that e ( + L) e ( ), while from Figure 3(a) we ee that e( + ) e( ) for all L. Therefore the econd variation i nonnegative for any δθ and δ, which implie that the correponding vertical olution i table. The tability of olution on the part of the vertical branch correponding to between y d (L) and d i le readily analyzed. However the problem can be tackled
X 1 X a X b X c X 2 Figure 4: Unit cell of the ubtrate. The projection of the flexible heet onto the ubtrate are labeled for variou initial poition. numerically by finding the mallet value of λ that guarantee the exitence of a nontrivial olution to the Euler-Lagrange equation βδθ + (e ( + ) e ( )) δθ = (7) for the functional G defined in (6) when δ =. Here δθ atifie the natural boundary condition δθ (L) = δθ () =, (8) and λ and are related via (5). We dicretize the econd derivative term in (8) by uing central difference and find the mallet value of λ > for which the determinant of the matrix of coefficient vanihe. Thi procedure yield a bifurcation point lightly larger than y d (L). 4 Atomitic Modeling To qualitatively verify the reult of the continuum modeling, we ue Accelry Ceriu 2 oftware to create an atomitic model of our ytem. We fix everal row of atom at the top edge of the flexible heet and move them toward the ubtrate b.1 nm per tep. After each tep, we minimize the energy and find the applied force by uing the tandard relationhip f = e. We do not tudy the influence of relative orientation of the ubtrate lattice and the flexible heet lattice and aume that thee orientation remain fixed (Figure 4). However we invetigate the effect of placing the flexible heet at different initial poition with repect to the ubtrate unit cell. Figure 5(a) how the dependence of the total energy on the diplacement of the top edge of the flexible heet from it initial poition. The tability lo occur at the point of rapid drop of the energy with the larget drop correponding to the onet of deformation in a perfectly vertical heet. Notice the trong dependence of the tability threhold on the initial placement of the flexible heet thi effect i lot in our continuum model, which doe not take into account poition of individual atom. Another effect that cannot be decribed within the ame continuum theory i hown in Figure 5(b). A the load i being increaed, the ucceive configuration eem to nap from one equilibrium to the next in order to conform to the exiting atomic tructure of the ubtrate. Thu the comparion with the atomitic model demontrate the need to incorporate the individual atomic poition into the continuum decription. Acknowledgment Thi work wa upported by NSF Grant DMS-4-7361 and DMS-3-5422.
56 54 52 x1 xa xb xc x2 4 3 2 5 1 48 1 2 3 4 42 44 46 48 5 52 54 Figure 5: (a) Total energy a a function of initial poition of graphene heet. (b) Succeive configuration of the heet during incremental loading. Reference [1] H. C. Schniepp, J. Li, M. J. McAlliter, H. Sai, M. Herrera-Alono, D. H. Adamon, R. K. Prud homme, R. Car, D. A. Saville, and I. A. Akay. Functionalized ingle graphene heet derived from plitting graphite oxide. The Journal of Phyical Chemitry Letter B, 11:8535 8539, 26. [2] A.M. Affoune, B. L. V. Praad, H. Sato, T. Enoki, Y. Kaburagi, and Y. Hihiyama. Experimental evidence of a ingle nano-graphene. Chemical Phyic Letter, 348:17 2, 21. [3] S. Horiuchi, T. Gotou, M. Fujiwara, T. Aaka, T. Yokoawa, and Y. Matui. Single graphene heet detected in a carbon nanofilm. Applied Phyic Letter, 84(13):243 245, 24. [4] J. Wang, M. Zhu, R. A. Outlaw, X. Zhao, D. M. Mano, and B. C. Holloway. Synthei of carbon nanoheet by inductively coupled radio-frequency plama enhanced chemical vapor depoition. Carbon, 42:2867 2872, 24. [5] M.-F. Yu. Fundamental mechanical propertie of carbon nanotube: Current undertanding and the related experimental tudie. J. of Eng. Mat. and Tech., 126:271 278, 24. [6] X.-S. Wang. Nanoparticle, nanorod, and other nanotructure aembled on inert urface. In Molecular Building Block for Nanotechnology. Springer, 27. [7] M.T. Cubere. Ultraonic machining at the nanometer cale. J. of Phy.: Conference Serie, 61(1):219 223, 27. [8] J. Zhou, Z. Shen, S. Hou, X. Zhao, Z. Xue, and Z. Shi, Z. Gu. Adorption and manipulation of carbon onion on highly oriented pyrolytic graphite tudied with atomic force microcopy. Appl. Surf. Sci., 253:3237 3241, 27. [9] X. Lu, M. Yu, H. Huang, and R. Ruoff. Tailoring graphite with the goal of achieving ingle heet. Nanotechnology, 1:269 272, 1999. [1] J.D. LeGrange. Microcopic manipulation of material by atomic force microcopy. Biophy. J., 64:93 94, 1993. [11] J.P. Wilber, A. Buldum, C.B. Clemon, D.D. Quinn, and G.W. Young. Continuum and atomitic modeling of interacting graphene layer. Phy. Rev. B, 75:45418, 27. [12] M. Arroyo and T. Belytchko. Finite crytal elaticity of carbon nanotube baed on the exponential Cauchy-Born rule. Phy. Rev. B, 69:115415 1 11, 24. [13] S. S. Antman. Nonlinear Problem of Elaticity. Springer, 25.