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

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Results

#### 3.1. Interatomic Forces Obtained at Atomically Specific Sites

#### 3.2. Quantitative Evaluation of Out-of-Plane Elastic Stiffness

## 4. Discussions

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Atomic-resolution AFM topography of a single-walled carbon nanotube (SWNT) with a well-defined radius ${\mathrm{R}}_{\mathrm{o}}=6.9\pm 0.1\text{}\AA $ in 3D view. The image size is $3\times 3\times 1\text{}{\mathrm{nm}}^{3}$. The atomically resolved depressing and protruding features are observed on the topmost part of a convexly curved surface.

**Figure 2.**(

**a**) Schematic illustration of a 3D-FFS measurement over the topmost $1\times 1\text{}{\mathrm{nm}}^{2}$ area of the SWNT as depicted in (

**b**). The upper (3D) and lower (2D) images in (a) are the surface topographies obtained before and during the measurement, respectively. The elongation of the honeycomb structure (dotted line) can be explained geometrically by assuming a point-like tip scanning across a curved surface (see text). (

**c**) 2D force field map obtained along the line indicated by a pair of counter arrows in (a). (

**d**) Force vs. distance relationships along the dashed lines in (c), corresponding the C1, C2, and H sites marked in (a) and (c) (for details see text).

**Figure 3.**(

**a**) Interatomic force ${F}_{\mathrm{int}}$ vs. vertical distance $z$ relationship at atomically specific C and H sites over the topmost part of convexly curved graphene. The C and H sites are respectively corresponding to the atomically specific sites representing depressing and protruding features in the topography image (i.e., the lower panel in Figure 2a) obtained during the 3D-FFS measurement. The interatomic forces ${F}_{\mathrm{int}}$ obtained at those sites were respectively averaged and plotted with the simulation curves (solid lines). (

**b**) The relationship between the simulation curves of solid and dashed lines obtained with and without taking relaxation into account, respectively. Those simulation curves are drawn together with the experimental-data plots obtained at the C site. The out-of-plane relaxation of the closest carbon atom due to the interatomic attractive force acting to the tip-apex atom is schematically illustrated in the inset of (b).

**Figure 4.**Quantitative relationship of (

**a**) out-of-plane elastic stiffness ${k}_{\perp}$ and (

**b**) elastic modulus ${E}_{\perp}$ at the atomically specific C and H sites of the convexly curved graphene with the relaxed displacement ${\mathrm{z}}_{\mathrm{rlx}}$ in the vertical direction due to tip-sample attractive interaction. The effective area of the interatomic forces acting between the tip-apex atom and the C and H sites are schematically illustrated in the inset of (b).

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**MDPI and ACS Style**

Ashino, M.; Wiesendanger, R.
Atomic-Site-Specific Analysis on Out-of-Plane Elasticity of Convexly Curved Graphene and Its Relationship to *Crystals* **2018**, *8*, 102.
https://doi.org/10.3390/cryst8020102

**AMA Style**

Ashino M, Wiesendanger R.
Atomic-Site-Specific Analysis on Out-of-Plane Elasticity of Convexly Curved Graphene and Its Relationship to *Crystals*. 2018; 8(2):102.
https://doi.org/10.3390/cryst8020102

**Chicago/Turabian Style**

Ashino, Makoto, and Roland Wiesendanger.
2018. "Atomic-Site-Specific Analysis on Out-of-Plane Elasticity of Convexly Curved Graphene and Its Relationship to *Crystals* 8, no. 2: 102.
https://doi.org/10.3390/cryst8020102