Elastic Properties and Stability of Physisorbed Graphene

Abstract

Graphene is an ultimate membrane that mixes both flexibility and mechanical strength, together with many other remarkable properties. A good knowledge of the elastic properties of graphene is prerequisite to any practical application of it in nanoscopic devices. Although this two-dimensional material is only one atom thick, continuous-medium elasticity can be applied as long as the deformations vary slowly on the atomic scale and provided suitable parameters are used. The present paper aims to be a critical review on this topic that does not assume a specific pre-knowledge of graphene physics. The basis for the paper is the classical Kirchhoff-Love plate theory. It demands a few parameters that can be addressed from many points of view and fitted to independent experimental data. The parameters can also be estimated by electronic structure calculations. Although coming from diverse backgrounds, most of the available data provide a rather coherent picture that gives a good degree of confidence in the classical description of graphene elasticity. The theory can than be used to estimate, e.g., the buckling limit of graphene bound to a substrate. It can also predict the size above which a scrolled graphene sheet will never spontaneously unroll in free space.

Graphene as a Prototypical Model for Two-Dimensional Continuous Mechanics

“…We note the excellent agreement with the excess energy reported for the zigzag edge of graphene (half the breaking energy G), λ = 1.7 nJ/m [18]. Using this value of λ together with C = 350 N/m [5] yields K c = 1.1 × 10 −3 Nm −1/2 , in fair agreement with the experimental value reported above for monolayer graphene.…”

“…[5], the values listed in Table 2 are obtained. The result for graphite agrees remarkably well with available experimental data [27,58].…”

“…The gr-gr interlayer interaction is about a factor of two larger in a scroll than in a single fold, because there is an interaction on both faces of graphene in most part of the scroll. Then, Van der Waals interaction favors the scroll over the fold, the more so that the bending energy in a scroll is not a strong penalty compared to a folded sheet [5,43].…”

“…Equations (8) and (10) (11) with C 33 the elastic constant of graphite along the c axis, where A = aζ 4 , B = bζ 10 , knowing that B/A = h 6 and C 33 = 6N A/h 5 [5].…”

“…Equation (1) relies on four elastic constants: the extensional stiffness A (N/m), the in-plane Poisson coefficient ν, the bending stiffness or flexural rigidity D (J), and the "bending Poisson coefficient" ν (see Section 6). In a previous paper [5], it was shown how experimental measurements together with atomistic calculations could provide reliable values of the elastic constants (A = 365 N/m, ν = 0.20, D = 1.6 eV). These values are in line with recent measurements of in-plane elastic constants of graphene: C ≡ A(1 − ν 2 ) = 342 N/m (2D Young modulus) and ν = 0.19 [6].…”