Determing the Elasticitiy of Graphene

Stiffness Measuring Using Atomic Force Microscope

  • Fig.1: (a) Scanning electron microscopy image of a large size graphene on an array of circular holes. Scale bar is 3 mm. (b) Schematic of the nanoindentation in free standing graphene membrane, (b) loading and unloading curves from the indentation with theory fitting from Ref. [6]. Reprinted with permission from AAAS. (c) Schematic of the graphene blaster test.Fig.1: (a) Scanning electron microscopy image of a large size graphene on an array of circular holes. Scale bar is 3 mm. (b) Schematic of the nanoindentation in free standing graphene membrane, (b) loading and unloading curves from the indentation with theory fitting from Ref. [6]. Reprinted with permission from AAAS. (c) Schematic of the graphene blaster test.
  • Fig.1: (a) Scanning electron microscopy image of a large size graphene on an array of circular holes. Scale bar is 3 mm. (b) Schematic of the nanoindentation in free standing graphene membrane, (b) loading and unloading curves from the indentation with theory fitting from Ref. [6]. Reprinted with permission from AAAS. (c) Schematic of the graphene blaster test.
  • Fig.2: Schematics of the force-displacement curve from AFM nanoindentation measurement.  Phase A denotes first step of the approaching process that the tip moves towards the sample but has not contacted. Phase B denotes second step of the approaching process that the tip deforms the sample and thus a repulsive force is measured. Phase C denotes the first step of the retraction curve that the tip move backwards the sample while the contact still remains. In this phase the point C* is the jump-off point and the maximum adhesion force is measured here. Phase D denotes the last step of the retraction curve that the tip moves backwards the sample without contact.
  • Fig.3: CVD graphene on SiO2 substrate (a) AFM height image and (b) reconstructed stiffness mapping.
Atomic force microscopy based nanoindentation technique provides a powerful approach to determine the elasticity of graphene. From the recorded force-displacement response, free standing graphene shows stiffness of 1TPa, which is the strongest material ever measured. In our work,the elasticity of graphene on substrate is investigated and the result shows system stiffness of 100 GPa, which indicates the role of graphene as a ultrathin surface reinforcing layer.
 
Introduction
As one of the most extraordinary materials, graphene has attracted considerable interest in many fields and its properties such as mechanical properties have become one of the current research hotspots [1]. A large number of experiments have been carried out and among them, the atomic force microscopy (AFM) based nanoindentation technique shows its uniqueness in determining the nanomechanical properties of graphene. AFM is a relatively new but powerful technique to locally determine the elasticity of materials, in which the AFM tip (usually made of Silicon) is employed to perform the nanoscale indentation and obtain the force-distance relation between the tip and the sample [2-3]. The recorded force-distance curve can be fitted through contact mechanics models, e.g. Derjaguin-Muller-Toropov (DMT) [4] model and Sneddon model [5], to extract the Young’s modulus.
 
Stiffness Measurement on Free Standing Graphene 
The outstanding mechanical properties of graphene have spurred scientists’ great enthusiasm for the application as an individual material or as a reinforcing agent in composites. The main mechanism for the exceptional mechanical properties of graphene comes from the extreme stability of the sp2 carbon-carbon bonds, which consist of the hexagonal lattice and opposes a variety of deformations. Lee and coworkers firstly reported the measurements on the mechanical properties of free standing graphene by the using AFM nanoindentation method (fig. 1) [6], and claimed that graphene is the “strongest material ever measured”. In their work, the force-displacement curve from the indentation process was recorded and the elastic properties of graphene can be extracted from the following equation:
 

where E2D is the 2D elastic stiffness, R is the tip radius and F is the applied force.

Therefore, an experimental value of the elastic stiffness was obtained as E2D = 340 ± 50 N/m, which corresponds to a Young’s modulus of 1.0 ± 0.1 TPa considering the effective thickness of 0.335 nm of graphene. In addition to the AFM technique, the blaster test is another straightforward way to obtain the elasticity of graphene as shown in figure 1c [7]. The deformation of the graphene membrane can be expressed by using Hencky’s solution for the geometrically nonlinear response of a clamped isotropic circular elastic membrane subjected to a pressure difference Δp across the membrane:

 
Where E is Young’s modulus, ν is Poisson’s ratio, t is the membrane thickness, K(ν) is a coefficient and a is a blister radius. The obtained Young’s modulus of graphene from the test is E = 1TPa, which is in line with value found by the AFM method.
 
Stiffness Measurement on Supported Graphene
The system elasticity of graphene on substrate, such as graphene on SiO2, is a fundamental parameter which is closely related to the performance of the system, while it is rarely reported. To address this concern, AFM based nanoindentation is employed here on the chemical vapor deposited (CVD) graphene on SiO2 substrate and figure 2 illustrates the schematics of the force-displacement curve from the AFM nanoindentation. The reduced elastic modulus E* can be extracted through DMT contact mechanics fitting and expressed as [4-6]:
 
where ΔF denotes the force difference at the deformation of ΔT and R is tip contact radius. Due to the thin thickness of the graphene layer, the deformation happens in both the tip and the sample, so the obtained E* becomes the system elasticity:
 
 
Where E and ν represent the elastic modulus and Poisson’s ratio, respectively. Figure 3a shows the obtained AFM height image of the CVD graphene on SiO2, where the grain boundaries are clear in the image. Figure 3b is the reconstructed Young’s modulus mapping of the sample Esample (256 ∙ 256 nanoindentations), from which a value of Esample = 100 GPa is obtained. Considering the intrinsic stiffness of SiO2 (76 GPa), this enhanced modulus of the system is mainly attributed to the surface reinforcing effect of the graphene due to its higher stiffness. Moreover, The elastic modulus of the top graphene layer can also be extracted through a thin film on subtract model descripted by Doerner and Nix [8]
 
where the parameter Φ is a weight function and can be expressed as: Φ = 1-e-βh/x. β is a constant and h/x is the ratio of the thickness h and indentation depth x, and E* is the measured reduced elastic modulus of the system. From this relation, the graphene elastic modulus of 900 GPa is found which is consistent with reported CVD graphene values. In same way, except from the elastic modulus, other physics parameter such as adhesion can also be extracted from this AFM nanoindentation measurement and superior adhesion property of nanostructures such as the graphene nanoscroll was reported [9].
 
Conclusion
In this work, we have reviewed the characterization of the mechanical properties of graphene using atomic force microscope based nanoindentation technique and also have proposed a methodology to measure the elastic modulus of graphene on substrate using AFM PF-QNM technique. The result indicates a system stiffness of 100 GPa, which indicates the surface reinforcing agency of the top graphene layer. In addition, the large number of the measurements gives more statistical values on the stiffness and also can be used to reconstruct to a stiffness map based on the modulus differences, which gives more intuitive and visual information of the sample. Therefore, this methodology has the potential to be used in the determination of elasticity in a variety of materials.
 
Acknowledgement
This work is supported by the Knut and Alice Wallenberg (KAW) and the Swedish National Science Council (VR 2016-05259). Mr. Sylvester Wambua Makumi, Mr. Tianbo Duan and Dr. Ishtiaq Wani are also acknowledged for the technical support on AFM measurements.
 

Authors
Hu Li1 and Klaus Leifer1

Affiliation
1Applied Materials Sciences, Department of Engineering Sciences, Angstrom Laboratory, Uppsala University, Sweden

Contact
Prof. Klaus Leifer
Applied Materials Sciences,
Department of Engineering Sciences,
Angstrom Laboratory,
Uppsala University, Sweden
Klaus.Leifer@angstrom.uu.se
https://www.teknik.uu.se/applied-materials-science/research-groups/electron-microscopy-and-nanoengineering/
 

References
[1] A.K. Geim, K.S. Novoselov: The rise of graphene, Nat. Mater. 6: 183–191 (2007) doi:10.1038/nmat1849.
[2] Fengzhen Sun, Hu Li, Klaus Leifer, Kristofer E. Gamstedt: Rate effects on localized shear deformation during nanosectioning of an amorphous thermoplastic polymer, Int. J. Solids Struct. 129: 40–48 (2017) doi:10.1016/j.ijsolstr.2017.09.016.
[3] Fengzhen Sun, Hu Li, Klaus Leifer, Kristofer E. Gamstedt: Effect of nanosectioning on surface features and stiffness of an amorphous glassy polymer, Polym. Eng. Sci. 58 1849–1857 (2018) doi:10.1002/pen.24793.
[4] B. Derjaguin, V. Muller, Y. Toporov: Effect of contact deformations on the adhesion of particles: J. Colloid Interface Sci. 53: 314–326 (1975) doi:10.1016/0021-9797(75)90018-1.
[5] Ian N. Sneddon: The relation between load and penetration in the axisymmetric boussinesq problem for a punch of arbitrary profile: Int. J. Eng. Sci. 3: 47–57 (1965)  doi:10.1016/0020-7225(65)90019-4.
[6] C. Lee, X. Wei, J.W. Kysar, J. Hone: Measurement of the Elastic Properties and Intrinsic Strength of Monolayer Graphene, Science 321 (5887) 385-388 (2008) doi:10.1126/science.1157996.
[7] S.P. Koenig, N.G. Boddeti, M.L. Dunn, J.S. Bunch: Ultrastrong adhesion of graphene membranes, Nat. Nanotechnol. (6) 543–546 (2011) doi:10.1038/nnano.2011.123.
[8] M.F. Doerner, W.D. Nix: A method for interpreting the data from depth-sensing indentation instruments, J. Mater. Res. (1) 601–609 (1986) doi:10.1557/JMR.1986.0601.
[9] Hu Li, Raffaello Papadakis, S. Hassan M. Jafri, Thomas Thersleff, Johann Michler, Henrik Ottosson, Klaus Leifer: Superior adhesion of graphene nanoscrolls, Commun. Phys. 1(44) (2018) doi:10.1038/s42005-018-0043-2.

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