Nanoindentation experiments for singlelayer rectangular graphene films: a molecular dynamics study
 Weidong Wang^{1}Email author,
 Shuai Li^{1},
 Jiaojiao Min^{1},
 Chenglong Yi^{1},
 Yongjie Zhan^{2} and
 Minglin Li^{3}
DOI: 10.1186/1556276X941
© Wang et al.; licensee Springer. 2014
Received: 9 October 2013
Accepted: 14 January 2014
Published: 22 January 2014
The Erratum to this article has been published in Nanoscale Research Letters 2014 9:228
Abstract
A molecular dynamics study on nanoindentation experiments is carried out for some singlelayer rectangular graphene films with four edges clamped. Typical load–displacement curves are obtained, and the effects of various factors including indenter radii, loading speeds, and aspect ratios of the graphene film on the simulation results are discussed. A formula describing the relationship between the load and indentation depth is obtained according to the molecular dynamics simulation results. Young’s modulus and the strength of the singlelayer graphene film are measured as about 1.0 TPa and 200 GPa, respectively. It is found that the graphene film ruptured in the central point at a critical indentation depth. The deformation mechanisms and dislocation activities are discussed in detail during the loadingunloadingreloading process. It is observed from the simulation results that once the loading speed is larger than the critical loading speed, the maximum force exerted on the graphene film increases and the critical indentation depth decreases with the increase of the loading speed.
Keywords
Molecular dynamics simulation Nanoindentation Rectangular graphene film Young’s modulus StrengthBackground
The perfect graphene is a kind of carbonaceous material which consists of twodimensional honeycomb lattice structures of single layer of carbon atoms. It is the basic unit to build other dimensional carbonaceous materials, such as zerodimensional fullerenes, onedimensional carbon nanotubes, and threedimensional graphite[1, 2]. Graphene sheets/ribbons/films have attracted the interest of the scientific community because of recent exciting experimental results[3–6]. Their growth, atomic makeup, electronics, doping, and intercalation have attracted many investigations[7–10]. A suspended graphene sheet[1, 11] can be used in a variety of ways, such as for pressure sensors or gas detectors[12] or mechanical resonators[13].
It is still debatable whether a graphene sheet is truly a twodimensional structure or if it should be regarded as a threedimensional structure since it exhibits a natural tendency to ripple, as observed in recent experiments[2, 14–16]. Carlsson addressed that an understanding of the coupling behaviors between bending and stretching of graphene sheets is necessary to fully explain the intrinsic ripples in a graphene sheet[15]. In addition to theoretical investigations, recent research has been carried out to measure the mechanical properties of suspended graphene sheets by utilizing an atomic force microscope (AFM)[17]. Through weak van der Waals forces, graphene sheets were suspended over silicon dioxide cavities where an AFM tip was probed to test its mechanical properties. Their Young’s modulus differs from that of bulk graphite. Poot and van der Zan[18] measured the nanomechanical properties of graphene sheets suspended over circular holes by using an AFM and suggested that graphene sheets can sustain very large bending and stretching prior to the occurrence of fracture, which indicates that the classical Kirchhoff plate theory used in the bending and vibration analysis of graphene sheets may not be suitable since deflection and stretching are considerable[19]. Some researchers thought that the large deflection plate theory of von Kármán may be a better candidate to model the graphene sheet, and they have characterized its bending and stretching through that theory[20, 21]. Lee et al. measured Young’s modulus and the maximum stress of graphene by using an AFM in the nanoindentation experiment[22] and reported the effect of grain boundaries on the measurement of chemical vapordeposited graphene[23]. Fang et al.[24] has studied the mechanical behavior of a rectangular graphene film under various indentation depths, velocities, and temperatures using molecular dynamics (MD) simulations. The physical models of the rectangular graphene film established by Fang et al. are doubly clamped using a bridgetype support and are loaded by a flatbottomed diamond tip.
Although the above research has been carried out for mechanical properties of graphene, the variation of the values of elastic modulus has not yet reached a consensus. Especially for rectangular graphene films, the relationship between the load and the indentation depth is not clear. Furthermore, there are few papers available which describe the deformation mechanisms and dislocation activities of graphene film during the nanoindentation processes in detail. These investigations are concentrated on tension deformation[25–28] and shear deformation[29]. Almost all of the available literatures on dislocation activities in graphene focus on theoretical studies and numerical simulations, including density functional theory (DFT)[26], tightbinding molecular dynamics (TBMD)[30], ab initio total energy calculation[30], and quantum mechanical computations[31]. Researchers always artificially applied defects or dislocations and then studied their effects on the properties and activities in graphene. However, due to the bottleneck of experimental study at nanoscale, a very few experimental observations of dislocation activities are available at present. Warner et al.[32] also reported the observation of dislocation pairs through HRTEM experiments and gave five possible mechanisms that describe how these dislocation pairs could have formed, namely, during the CVD growth, electron beam sputtering of carbon dimers along a zigzag lattice direction, from surface adatom incorporation, from a monovacancy, and from a StoneWales defect. They then concluded that edge dislocations result in substantial deformation of the atomic structure of graphene, with bond compression or elongation of ±27%, plus shear strain and lattice rotations.
In this article, some MD simulations of nanoindentation experiments are performed on a set of singlelayer rectangular graphene films with four clamped edges. The dislocation activities and the deformation mechanism are discussed, and a formula is introduced in order to describe the relationship of load and indentation depth and to measure the mechanical properties of graphene.
Methods
When performing MD simulations, we use the canonical (i.e., NVT) ensemble and control the temperatures at an ideal temperature of 0.01 K. In order to avoid the complex effects of the atomic thermal fluctuations, the temperature is regulated with the NoséHoover method and the time step was set to 1 fs. During the simulation, one key step, named energy minimization and relaxation, should be carried out to make the system remain in the equilibrium state with lowest energy. Then, the indentation experiment was executed and the simulation results were output for further research.
Results and discussion
Loading and unloading properties
We take the case of the graphene film with an aspect ratio of 1.2 and the diamond indenter with a radius of 2 nm as an example to describe the indentation experiment in the following. The indenter was placed over the geometric center of the graphene film and forced to move in the direction perpendicular to the original graphene surface. Figure 1 gives the atomic configurations of the system model during the indentation experiment at a speed of 0.20 Å/ps. The atoms on the edge of the graphene film remained in a static state due to fixed boundary conditions. After enough loading time, the graphene film is eventually pierced through by the indenter, appearing some fractured graphene lattices. The load–displacement curves can be attained from the data of intender load (F) and indentation depth (d) calculated in MD simulations. The moment the load–displacement curve drops suddenly is considered to be a critical moment. In our simulations, the load suddenly decreased once the indentation depth exceeded 5.595 nm, defined as the critical indentation depth (d_{c}), and the corresponding maximum load (F_{max}) is 655.08 nN.

Stage I. The unloading process is done before the indentation depth reaches the critical depth, d_{c}. The graphene sheet almost can make a complete recovery, i.e., restore its initial structures, and the curves of reloading processes almost perfectly match the initial loading curve while the unloading curve shows very small deviations from the initial one, as shown in the inset of Figure 3. In general, the almostperfect coincidence is due to the fact that the carbon covalent bonds and the graphene lattice structure are not destroyed. It can be concluded that there is no plastic deformation in this stage, i.e., the graphene undergoes elastic deformation.

Stage II, i.e., the yellow region in Figure 3. In Figure 4a,b,c, it is after the indentation depth exceeded d_{c} that the unloading process begins. The graphene sheet cannot make a complete recovery, and there exited broken covalent bonds after the unloading process. In the reloading process, the maximum force exerting on graphene is much smaller than that in Figure 3, which denotes the fracture of graphene lattices. Figure 4b describes the state where the unloading process begins, and Figure 4c describes the state where the unloading process ends. After the loading process, there exited broken bonds and fractured lattices in the middle of the graphene film and these defective structures did not recover during the unloading process. Therefore, the deformation of the graphene described in this figure can be considered as a plastic type.
Young’s modulus and strength of the graphene film
where E^{2D} is the 2D elastic modulus, i.e., Young’s modulus, of the single layer graphene film. The strain energy density of graphene, as a standard 2D material, can be represented by the energy of per unit area. Then, the corresponding pretension and elastic modulus can be expressed as σ_{0}^{2D} and E^{2D}, respectively, with the unit N/m. The common pretension and elastic modulus of a 3D bulk material can be obtained through these 2D values divided respectively by the effective thickness which is always treated as the layer spacing of the graphite crystal, i.e., 3.35 Å. q is an nondimensional value, q = 1/(1.05  0.15ν  0.16ν^{2}) = 0.9795, where ν denotes Poisson’s ratio, ν = 0.165[3, 18, 21]. It is reported that when r/R > 0.1, the indenter radius has a significant influence on the load–displacement properties[38, 39]. In our simulations, r/R > 0.1; thus, Equations 2 and 3 are corrected by a factor of (r/R)^{3/4} and (r/R)^{1/4}, respectively.
Mechanical properties of the singlelayer graphene film from nanoindentation experiments
Indenter radius (Å)/speed (Å/ps)  2D elastic modulus (N/m)  3D elastic modulus (TPa)  2D pretension (N/m)  3D pretension (GPa)  2D max stress (N/m)  3D max stress (GPa) 

10/0.10  375.0644  1.1196  38.8546  115.9840  72.4895  216.3866 
10/0.20  375.0096  1.1194  38.8589  115.9966  72.4771  216.3496 
20/0.10  335.0012  1.0000  28.5092  85.1021  66.1326  197.4106 
20/0.20  335.2572  1.0008  28.4879  85.0385  66.0994  197.3115 
30/0.10  349.1828  1.0423  22.7998  68.0590  67.4504  201.3445 
30/0.20  348.8383  1.0413  23.0197  68.7154  67.6680  201.9940 
Average  353.0589  1.0539  /  /  68.7195  205.1328 
Other parameters’ influences on nanoindentation experiments
Conclusions
Some MD simulations of nanoindentation experiments on singlelayer rectangular graphene sheets have been carried out in order to obtain the mechanical properties of graphene. A correlation between the load and the indentation depth is constructed, and Young’s modulus and the strength of graphene are obtained in the end. The simulation results show that the unloaded graphene film could make a complete recovery if the maximum indentation depth is less than the critical indentation depth, and the graphene film undergoes elastic deformation during the whole loadingunloadingreloading process. However, if the maximum indentation depth is larger than the critical indentation depth, the graphene sheet could not restore its original structures after unloading and the graphene deforms plastically. Based on the simulations of nanoindentation at different loading speeds and indenter radii, it can be observed that the maximum load increased and the critical indentation depth decreased with the increase of loading speed. In addition, the indenter radius has a remarkable influence on the forcedisplacement curve. As the indenter radius increases, the critical load and the critical indentation depth also increase.
Notes
Abbreviations
 2D:

twodimensional
 AFM:

atomic force microscope
 AIREBO:

adaptive intermolecular reactive empirical bond order
 GPa:

gigapascals
 MD:

molecular dynamics
 NVT:

constant number of particles, volume, and temperature
 SW:

StoneWales 5775
 TPa:

terapascals.
Declarations
Acknowledgements
We acknowledge the financial support provided by the Fundamental Research Funds for the Natural Science Basic Research Plan in Shaanxi Province of China (grant no. 2013JM7017), subsequently by the National Natural Science Foundations of China (grant no. 51205302 and no. 50903017) and the Central Universities in Xidian University (grant no. K5051304006). We also would like to thank all the reviewers for their comments and kind suggestions to our manuscript and all the editors for their careful corrections on the final version of the article.
Authors’ Affiliations
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