Attractive force-driven superhardening of graphene membranes as a pin-point breaking of continuum mechanics

Abstract

Bending at the nanometre scale can substantially modify the mechanical, chemical and electronic properties of graphene membranes. The subsequent response of chemical bonds leads to deviations from plate idealisation in continuum mechanics. However, those phenomena have thus far been investigated exclusively by measuring the electronic properties of graphene deformed by compressing and stretching with local-probe techniques. Here, we report that the interatomic-attractive forces applied on the convexly-curved graphene by the probe tip give rise to a pin-point breaking of the plate idealisation in the continuum mechanics, facilitating atomically-localised enhancements in its chemical reactivity and mechanical strength. Thorough characterisations were conducted by atomic force microscopy and force field spectroscopy on hollow nanotubes, rolled-up graphene, with different diameters. Their topmost parts supplied well-defined curvatures of the convex graphene. We found that a significant enhancement in the out-of-plane Young’s modulus from 13 to 163 GPa, “superhardening”, was realised with the nonlinear transition of bond configurations. Our findings provide a fundamental understanding of the relationships between the structure of atomistic membranes and the dynamic behaviour of approaching exterior atoms or molecules and their subsequent interplay with chemical and mechanical properties. Thus, these results encourage the application of such membranes in functionally-controllable materials or devices.

Atomic-Site-Specific Analysis on Out-of-Plane Elasticity of Convexly Curved Graphene and Its Relationship to s p 2 to s p 3 Re-Hybridizatio

“…The 3D damping fields specify the locations where the tip apex inelastically interacts with the sample surface. As described in [33], such inelastic signals are completely absent in the 3D damping fields, meaning that the interatomic interactions are purely elastic over the central ridge of the convexly curved graphene. Consequently, the int ( ) plots in Figure 3(a) represent atomically specific features in its elasticity.…”

“…On the other hand, as z rlx increases further, E ⊥ is found to be significantly enhanced (E max ⊥ > 100 GPa) for the C site, whereas E ⊥ for the H site still stays at a similar value. The increase of the relaxed displacement in the out-of-plane direction would make the carbon atom and its three nearest neighbors no longer planar but instead located in the corners of a pyramid [33]. Such pyramidalization is accounted for by the π-orbital axis vector (POAV) constructions [33,44].…”

“…The radius of the tip-apex curvature used in our study was estimated to be R t = 5.8 ± 0.6Å by a deconvolution method for the sections in 3D topographies [31]. The atomically resolved AFM imaging enables us not only to confirm surface cleanliness and to determine the chirality of curved graphene but also to characterize its chemical and mechanical properties at atomically specific sites [28,32,33]. Atomic-resolution AFM topography of a single-walled carbon nanotube (SWNT) with a well-defined radius R o = 6.9 ± 0.1 Å in 3D view.…”

“…The vertically sectional force map along the horizontal line, indicated by a pair of counter arrows, even represents an atomic-scale contrast of the total interaction forces total , as shown in Figure 2(c), where the positions corresponding to the C1, C2, and H sites in Figure 2 In three-dimensional (3D) force field spectroscopy (3D-FFS), tip-sample interaction forces are measured by recording ∆ f (x, y, z) and converting that to F(x, y, z). At every recording point (x, y) over the sample surface, the tip-sample distance z is regulated to attain the same ∆ f value and consequently to obtain the surface topography as in the case of previous imaging with atomic resolution [28,32,33]. Acquisition of the atomically resolved topography, which also took place during the 3D-FFS measurement, allows for an atomic-site-specific analysis of interaction forces.…”

“…because their interaction was found to be elastic, as mentioned above. The so-called "out-of-plane" elastic stiffness k ⊥ of the convexly curved graphene was expected to vary with the local increment of the curvature ∆(1/R) rlx , with a linear correlation to z rlx , as described in [33]. Since Hooke's law applies for small displacement ∆z rlx , the "out-of-plane" elastic stiffness k ⊥ can be derived from the infinitesimal force change dF int for the infinitesimal displacement dz rlx as k ⊥ = dF int /dz rlx .…”