Bending Ultrathin Graphene at the Margins of Continuum Mechanics

Deviations from continuum mechanics are always expected in nanoscale structures. We investigate the validity of the plate idealization of ultrathin graphene by gaining insight into the response of chemical bonds to bending deformations. In the monolayer, a bond orbital model reveals the breakdown of the plate phenomenology. In the multilayer, objective molecular dynamics simulations identify the validity margin and the role of discreteness in the plate idealization. Our result has implications for a broad class of phenomena where the monolayer easily curves, and for the design of mass and force detection devices.

Figure 1

(a) Diagram showing the $\pi$-orbital in planar graphene and its change into $h_{\pi }$ under bending. (b) Relative orientation of two $h_{\pi }$ orbitals located on a carbon-carbon bond of bent graphene. The planes delineated by the ${\sigma }_i$ bonds in (a) and by $h_{\pi }$ and ${\sigma }_i$ axes in (b) were hatched. The upwards arrows are the POAVs.

Figure 2

(a) A bent monolayer in the armchair direction is composed of primitive motifs containing only four atoms. Neighboring motifs are generated by translating and rotating this motif along and around the OZ axis. (b) An $N=3$ graphene bent with $\Omega =3.6\mathrm{deg}$ ($1/R=0.15{\mathrm{nm}}^{-1}$), as obtained by objective MD relaxations. Dashed line is the neutral surface. (c) Length distribution of the carbon-carbon bonds oriented along the principal curvature, when $\Omega =0.2\mathrm{deg}$.

Figure 3

(a) DFTB bending strain energy as a function of curvature squared, from monolayer to 5-layer graphene. (b) Second derivative of the bending energies extracted from the microscopic data, and D values (crosses) predicted by Eq. (7). Plate model for graphene based on microscopic data: (c) Young’s modulus and (d) thickness (normalized by $N$) as a function of $N$. The horizontal lines mark the $C/Z_0$ and $Z_0$ values.