An analytic investigation for the edge effect on mechanical properties of graphene nanoribbons
AAn analytic investigation for the edge effect on mechanical properties of graphenenanoribbons
Guang-Rong Han, Jia-Sheng Sun, and Jin-Wu JiangCitation: Journal of Applied Physics , 064301 (2018);View online: https://doi.org/10.1063/1.5012562View Table of Contents: http://aip.scitation.org/toc/jap/123/6Published by the American Institute of Physics
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Guang-Rong Han, Jia-Sheng Sun, and Jin-Wu Jiang a) Shanghai Institute of Applied Mathematics and Mechanics, Shanghai Key Laboratory of Mechanics in EnergyEngineering, Shanghai University, Shanghai 200072, People’s Republic of China (Received 7 November 2017; accepted 28 January 2018; published online 12 February 2018)We derive analytical expressions for the Young’s modulus and the Poisson’s ratio of the graphenenanoribbon, in which free edges are warped by the compressive edge stress. Our analyticalformulas explicitly illustrate the reduction of the Young’s modulus by the warped free edges,leading to the obvious width dependence for the Young’s modulus of the graphene nanoribbon.The Poisson’s ratio is also reduced by the warped free edges, and negative Poisson’s ratio can beachieved in the graphene nanoribbon with an ultra-narrow width. These results are comparablewith previous theoretical works.
Published by AIP Publishing. https://doi.org/10.1063/1.5012562
I. INTRODUCTION
Mechanical properties of graphene have been widely stud-ied by previous experimental and theoretical works. Earlierexperiments reported the Young’s modulus of bulk graphite tobe 1.06 In 2007, Frank et al. measured theYoung’s modulus of the few-layer graphene, and found that thevalue is approximately at 0.5 TPa. In 2008, the Young’s modu-lus of the monolayer graphene was measured to be 1 : et al. A huge number of theoretical works have alsobeen devoted to the calculation of the Young’s modulus forgraphene. Liu et al. performed density functional calculationsto investigate the mechanical properties of graphene, and foundthat the Young’s modulus of graphene is 1.05 TPa andthe Poisson ratio is 0.186. Shokrieh and Rafiee obtained themolecular potential energy and the Young’s modulus of thegraphene sheet is obtained by using the beam model. More dis-cussions on the mechanical properties of graphene can be foundin some recent review articles (see, for example, the study byNovoselov et al. and Akinwande et al. )As a result of its quasi-two-dimensional structure, thefree edges play an important role in various physical propertiesof the graphene nanoribbons (GNRs). In particular, the edgestress can cause strong effects on various mechanical proper-ties. The origin of the edge stress is related to the under coordi-nation of atoms at the free edge. The chemical bonds for theedge atoms will be different from bonds of the interior atoms,leading to compressive/tensile deformation of the free edge. Inpractice, the edge stress can be defined in analogy with the sur-face stress of a three-dimensional (3D) crystal. At the freeedges of the GNRs, the compressive edge stress can cause atransition from the planar configuration into the warped config-uration, which results from the buckling phenomenon induced by the extremely high in-plane stiffness (35 GPa) andultrasmall bending modulus (1.44 eV). It has been shown thatthe warping structure has strong effects on the elastic propertiesof GNRs. In 2016, Jiang and Park found the negativePoisson’s ratio in GNRs using an inclined plate model, where the value of the Poisson’s ratio is governed by the interplaybetween the width and the warping amplitude of the edge.Furthermore, Jiang also found that the three-dimensionalwarped structure with free edges will transform to the two-dimensional planner structure at the critical strain e ¼ : et al. investigated the effect of the warpededge on the buckling process of GNRs.In particular, the edge effects on the Young’s modulus andthe Poisson’s ratio have also been investigated extensively. Forinstance, Reddy et al. performed molecular dynamical (MD)simulations to demonstrate that the edges strongly affect theelastic properties of the graphene sheet when the width is lessthan 8 nm. Some studies have found that the Young’s modulusof the GNR increases with increasing width, resulting from thefree edges, while others found an opposite width depen-dence for the Young’s modulus. The edge induced widthdependence for the Poisson’s ratio has also been investigatedthrough some numerical methods.
Overall, various numerical approaches have been adoptedby most existing works to investigate the edge effects on theYoung’s modulus and the Poisson’s ratio of the GNRs. An ana-lytical study can explicitly disclose the relation between thefree edge and the Young’s modulus and the Poisson’s ratio,which is still lacking. We thus provide an analytical derivationto reveal the direct relation between the warped free edge andthe Young’s modulus and the Poisson’s ratio in the GNR.In this paper, we derive the analytical expression for theYoung’s modulus and the Poisson’s ratio of the GNRs, inwhich free edges are warped. The effect of the warped edgeis considered analytically by considering the elastic energyof the warped configuration. The Young’s modulus and thePoisson’s ratio of the GNRs both increase monotonicallywith the increase of the width. Our analytical results arecompared with the existing theoretical works.
II. ELASTIC ENERGY DENSITY
Free edges in the graphene nanoribbons (GNRs) arewarped by the compressive edge stress. The shape of thewarped edge can be described by a) Author to whom correspondence should be addressed: [email protected] , 064301-1
JOURNAL OF APPLIED PHYSICS , 064301 (2018) x ; y ð Þ ¼ Ae (cid:2) y = l c sin p x = k c ð Þ ; (1)where A is the amplitude of the ripple, and l c is the penetra-tion length. k c is the wave length of the warping ripple. Thez-axis is perpendicular to the graphene plane, while thex-axis is along the edge.We can compute the strain energy of the warped edge,i.e., the edge energy. To do so, we consider a semi-infinitesheet structure in the region (cid:2)1 < x < þ1 ; (cid:3) y < þ1 .The stresses are r ¼ r ¼ r ¼ r ¼ E þ (cid:2) ð Þ (cid:2) (cid:2) ð Þ (cid:2) (cid:2) ð Þ u þ (cid:2) u þ u ð Þ (cid:2) (cid:3) ; (2)where (cid:2) is the Poisson’s ratio and E is the Young’s modulus.From the boundary condition, r ¼
0, we obtain u ¼ (cid:2) (cid:2) (cid:2) ð Þ u þ u ð Þ : (3)From the expression of the warped configuration in Eq. (1),the strain in this structure can be obtained by u ij ¼ @ z @ x j @ z @ x i ,i.e., u ¼ @ z @ x (cid:4) (cid:5) u ¼ @ z @ y (cid:4) (cid:5) u ¼ (cid:2) (cid:2) (cid:2) ð Þ @ z @ x (cid:4) (cid:5) þ @ z @ y (cid:4) (cid:5) " u ¼ u ¼ : (4)As a result, the warping induced strain energy is U ¼ E þ (cid:2) ð Þ u ik þ (cid:2) (cid:2) (cid:2) u ll (cid:4) (cid:5) : (5)Using the exact expression for each strain in Eq. (4), weobtain U ¼ M @ z @ x (cid:4) (cid:5) þ @ z @ y (cid:4) (cid:5) " ; (6)where M ¼ E = ð (cid:2) (cid:2) Þ .The energy associated with the free edge is U ¼ s e u þ E e u ; (7)where s e is edge stress, which is determined by the bondingconfiguration of edge atoms and is thus a width independentconstant. E e is the edge Young’s modulus at the edge of theGNR.Using the explicit expressions for the strain in Eq. (4),we obtain U ¼ s e @ z x ; ð Þ @ x (cid:4) (cid:5) þ E e @ z x ; ð Þ @ x (cid:4) (cid:5) : (8)The total energy in one periodic length k c ¼ p = k for thewarped edge is U e ¼ ð p = kx ¼ U dx þ ð p = kx ¼ ð y ¼ U dxdy ! ¼ A p s e k c þ A p E e k c þ A M þ l c p k c þ l c p k c ! l c ; (9)where the prefactor of 2 is to consider a pair of opposite freeedges in the nanoribbon system.The amplitude of the warped edge will decrease grad-ually by applying the tensile strain. The dependence of thewarping amplitude on the tensile strain can be describedby A ¼ A cos p u e c (cid:6) (cid:7) , with A ¼ :
26 nm as the amplitude ofthe warped edge without strain. The warped edge will befattened by tensile strain above the critical strain e c ¼ : Substituting this function of A into Eq. (9),we obtain the strain dependence of the linear density forthe edge energy U e ¼ A p s e cos p u e c (cid:4) (cid:5) k c þ A p E e cos p u e c (cid:4) (cid:5) k c þ A M þ l c p k c þ l c p k c ! cos p u e c (cid:4) (cid:5) l c : (10)As a result, the edge energy per volume is U e ¼ A p s e cos p u e c (cid:4) (cid:5) k c Wh þ A p E e cos p u e c (cid:4) (cid:5) k c Wh þ A M k c þ l c p k c þ l c p k c ! cos p u e c (cid:4) (cid:5) l c Wh ; (11)where w and h are the width and the thickness of the system,respectively. We have introduced the quantity MM ¼ E (cid:2) (cid:2) : (12)In addition to the edge energy, there is the usual strainenergy in graphene U (cid:3) ¼ l u ik (cid:2) d ik u ll (cid:4) (cid:5) þ Ku ll ; (13)where K ¼ k þ l is the bulk modulus and k and l are the Lam (cid:2) e coefficients.The total energy of the GNRs is the summation of theedge and the strain energy , 064301 (2018) ¼ U e þ U (cid:3) ¼ l u ik (cid:2) d ik u ll (cid:4) (cid:5) þ Ku ll þ A p s e cos p u e c (cid:4) (cid:5) k c Wh þ A p E e cos p u e c (cid:4) (cid:5) k c Wh þ A M k c þ l c p k c þ l c p k c ! cos p u e c (cid:4) (cid:5) l c Wh : (14) III. EDGE EFFECTS ON THE YOUNG’S MODULUS ANDPOISSON’S RATIO
The stress tensor can be derived from its definition, r ik ¼ @ U @ u ik , which gives r ik ¼ Ku ll d ik þ l u ik (cid:2) u ll d ik (cid:4) (cid:5) (cid:2) k c Wh e c A p k c s e þ A p E e þ A M h (cid:4) p k c l c þ l c p k c þ k c p l c ! d ik u d ik ¼ Ku ll d ik þ l u ik (cid:2) u ll d ik (cid:4) (cid:5) þ Bu d ik ; (15)where we have introduced the parameter BB ¼(cid:2) k c Wh e c A p k c s e þ A p E e þ A M h (cid:4) p k c l c þ l c p k c þ k c p l c ! : (16)We thus can obtain the strain tensor u ik u ik ¼ d ik r ll K þ B ð Þ þ r ik (cid:2) K r ll þ B r K þ B d ik l : (17)To calculate the Young’s modulus and the Poisson’s ratio,we stretch the graphene along the x-direction with a tensionper area as p . The only nonzero component in the stress ten-sion is r xx ¼ p . According to Eq. (17), the nonzero compo-nents of the strain tensor are u xx ¼
12 2 K þ B ð Þ þ K l K þ B ð Þ (cid:8) (cid:9) pu yy ¼
12 2 K þ B ð Þ (cid:2) K þ B l K þ B ð Þ (cid:8) (cid:9) p : (18)The Young’s modulus and Poisson’s ratio are thus obtainedfrom their definitions E ¼ B þ K ð Þ l K þ l (cid:2) ¼ B þ K (cid:2) l K þ l ; (19)where the parameter B describes the edge effect as defined inEq. (16). Figure 1 shows the width dependence for the Young’smodulus and the Poisson’s ratio of the GNRs. We have usedthe following value for the parameters. The Young’s modulusof the pure graphene without edge is E ¼ :
05 TPa and thePoisson’s ratio (cid:2) ¼ : The critical strain e c ¼ :
7% isfrom Jiang. The other parameters for the edges of the gra-phene are from the original paper by Shenoy et al. The initialamplitude A ¼ :
26 nm, the penetration length l c ¼ k c ¼
10 nmare from the study by Shenoy et al. The edge stresses are s e ¼ : E e ¼ The standard Newton equations ofmotion are integrated by using the velocity Verlet algorithmwith a time step of 1 fs. The numerical results are in reason-able agreement with our analytical predictions, especially forGNRs of a large width. However, it should be noted thatthere are some deviations between the numerical results andthe analytical predictions for ultra-narrow GNRs, which shallbe attributed to the strong interplay between the two freeedges that was not considered in the analytical derivation.
FIG. 1. The width dependence for the Young’s modulus E = E in (a) andPoisson’s ratio (cid:2)=(cid:2) in (b) for GNRs. Values are with respect to the value ofpure graphene without free edges, i.e., E ¼ :
05 TPa and (cid:2) ¼ : Lines are analytical predictions, while points are MD results. , 064301 (2018) igure 1(a) shows the width dependence for the Young’smodulus. The Young’s modulus for the zigzag GNRs is smallerthan that of the armchair GNRs, but the difference is small. Forboth armchair and zigzag GNRs, the Young’s modulusincreases with increasing width, and will approach the value ofthe Young’s modulus for the pure graphene without free edges.It has also been found by Zhao et al. that the Young’s modu-lus for both armchair and zigzag GNRs increase with theincrease of width. Bu et al. also found that the Young’s mod-ulus increases along the width for narrower GNRs. The work ofLu et al. obtained an opposite result that the Young’s modu-lus decreases with the increasing of GNR width.Figure 1(b) shows the width dependence for the Poisson’sratio of the armchair and zigzag GNRs. The Poisson’s ratiofor both armchair and zigzag GNRs increase with the increaseof the width, and will saturate at the value of the Poisson’sratio in pure graphene without free edges. There is a small dif-ference in the Poisson’s ratio between the armchair and zigzagGNRs, and the Poisson’s ratio in the armchair GNR is slightlylarger than that of the zigzag GNR. Our analytical resultsagree with the predictions by Georgantzinos et al. , wherethe Poisson’s ratio also increases with increasing width. Ourresults are different from the work by Wang et al. , in whichthe Poisson’s ratio of armchair (zigzag) GNRs decreases(increases) with the increasing width.Figure 1(b) shows that, due to the edge effect, thePoisson’s ratio can be negative for ultra-narrow GNRs, wherethe warped edge takes dominant effects. The warped edges willbe effectively expanded during the flattening process underexternal stretching. This edge induced negative Poisson’s ratiophenomenon was also found by Jiang and Park. We note that there are similar size effects of 3D materi-als.
In particular, for the 2D ribbon of width w , we candivide the system into three regions, including one interiorregion of size w (cid:2) l c and two edge regions of size l c . Thepenetration length l c can be regarded as the size of the edgeregion. Simple algebra gives the effective properties for thewhole ribbon D ¼ D (cid:2) l c W ð D (cid:2) D e Þ ; (20)where D is the elastic properties like the Young’s modulusand Poisson’s ratio in this work, and D and D e are the corre-sponding mechanical quantities for the interior and edgeregions. We thus obtain a general formula for the widthdependence of the effective Young’s modulus and thePoisson’s ratio. The width dependence is reflected by theparameter B in Eq. (16), which is indeed inverse to the widthof the ribbon and thus agrees with the general argument here.For 3D materials, there are similar general arguments to Eq.(20). As a result, there are similar size effects in the 3D struc-tures. For example, Miller and Shenoy have shown that theelastic properties of nanosized 3D structural elements have asimilar size dependence. IV. CONCLUSION
To summarize, we have performed an analytical studyfor the effect of the warped edge on the mechanical properties for the graphene nanoribbon, including theYoung’s modulus and the Poisson’s ratio. The warped edgeeffect is considered analytically by using the expression ofthe elastic energy of the warped configuration. We obtain theanalytical expression disclosing the relation between theedge properties and the Young’s modulus and Poisson’sratio. More specifically, the Young’s modulus increases withthe increase of width, and the Poisson’s ratio also increaseswith increasing width. These results are comparable withprevious works.
ACKNOWLEDGMENTS
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