Theoretical Analysis of the Intrinsic Mechanical Properties of Graphene
Songül Güryel1, Balázs Hajgató1, Yves Dauphin2, Jean-Marie Blairon2, Hans Edouard Miltner2, Paul Geerlings1, Frank De Proft1 and Gregory Van Lier1
1Free
University of Brussels - Vrije Universiteit Brussel (VUB), Research Group General Chemistry (ALGC), Pleinlaan 2, B-1050 Brussels, Belgium
2SOLVAY
S.A., Innovation Center, Ransbeekstraat 310, B-1120 Brussels, Belgium
More info: www.nanoscience.be, [email protected], [email protected]
Introduction
The Young's modulus, E, can be calculated by dividing the tensile stress by the tensile strain, which
correspond to the force (F) required for this distortion over the original cross-sectional area through
which the force is applied (A0), and the relative elongation (ΔL over the original length L0) respectively
Graphene, defined as a monolayer of carbon atoms packed in a honeycomb lattice, has
recently gained significant attention. In particular, its excellent mechanical properties are an
important advantage for the practical applications of graphene. These mechanical properties
have extensively been investigated, and in particular, the Young’s Modulus has been
predicted using a range of experimental and theoretical approaches.
FL0
tensile stress  F A0
E
 

tensile strain  L L0 A0 L
In this study, the Young’s modulus of graphene has been investigated theoretically by using
semi-empirical PM6, and Density Functional Theory (DFT) with the PBE, HSE1PBE
functionals and the 6-31G* basis set. When external strain for monolayer graphene is
applied, the internal forces and the Young’s modulus are calculated using different
approaches for both finite and periodic systems[1]. These results are in a good agreement
with theoretical and experimental results from the literature[2,3]. In addition, the intrinsic
mechanical properties of bi-layer graphene is also analysed to show the influence of the
amount of layers and of the size of the graphene sheets on the intrinsic mechanical
properties.
In order to predict the area A0, the following approximation was used, whereby the cross-sectional area
of graphene was calculated as a sum of the Van der Waal Radius for two adjacent carbon atoms with
R = 1.70 Å (see reference 3), as depicted schematically below in Figure.
R; Van der Waals Radii, 1.70 Å
r; Covalent Radii, 0.77 Å
System III
System I
System II
System IV
C150H34
C332H40
 Graphene sheet model considered for the prediction of the Young’s modulus, showing the atoms that are pulled apart (left) and the resulting forces (right) for Systems I to IV.
 The structural deformation on the systems was shown by color coding the deviance of the angle from ideal sp2 hybridization angle.
Periodic Calculations
∆=
(120 − 𝛼1 )2 + 120 − α2
L/Å
c
d
Different graphene models used for periodic calculations with 2 carbon atoms (a), 4 carbon atoms (b), and 6 carbon atoms
(c) per unit cell (d) bi-layer for 6 carbon atoms.
The unit-cell size and shape dependence was analysed in the case of monolayer graphene. Three different unit cells were
considered, namely the minimal 2 carbon unit cell (a), the 4 carbon atom cell (b), and the 6 carbon atom one (c) (see Figure
above). Based on bond-lengths, all the 3 unit cells give the expected nearly degenerate carbon-carbon bond lengths. Additionally,
the supercell approach using 2x2 and 3x3 primitive unit cells (a, b, or c) per unit cell give almost similar results, therefore, no extra
periodicity was found for single layer graphene. The calculations were performed with different DFT functionals, namely with the
pure PBE and the hybrid (developed for solids) HSEh1PBE functionals. The two functionals give comparable results.
L/Å
2
PM6
System I
System II
L1=12.50 - 0.89 %
1.42
1.18
L2=12.75 - 2.9 %
1.39
1.01
L3=13.00 - 4.9 %
1.37
0.953
L4=13.25 - 6.9 %
1.35
0.911
L5=13.50 - 8.9 %
1.31
0.872
L6=13.75 - 11 %
1.25
0.831
L7=14.00 - 13 %
1.19
0.789
L8=14.37 – 16 %
1.06
0.726
L9=14.87 – 20 %
0.787
0.633
L/Å
b
+ 120 − α3
E / TPa
L0= 12.39
a
2
E / TPa
PM6
L0= 12.36
System III
System IV
L1= 12.42 - 0.5 %
1.86
1.24
L2=12.61 - 2.0 %
1.76
1.036
L3=12.98 - 5.0 %
1.65
0.943
L4=13.35 - 8.0 %
1.601
0.879
L5=13.84 – 12 %
1.501
0.784
L6=14.34 – 16 %
1.34
0.697
L7=14.83 – 20 %
1.069
0.613
Length (L in Å ) and the predicted Young’s modulus (E in TPa) of
graphene for system I, System II, System III and system IV.
Continuum Mechanical Simulation
E / TPa
PBE
HSEh1PBE
L1= 2.47 - 0.5 %
1.01
1.11
L2=2.51 - 2.0 %
0.969
1.07
L3=2.58 - 5.0 %
0.883
0.983
L4=2.65 - 8.0 %
0.807
0.906
L5=2.75 - 12 %
0.708
0.802
L6=2.85 - 16 %
0.613
0.706
L7=2.95 – 20 %
0.526
0.611
Displacement field
Strain field
If only the distance b is elongated, with the rest of the material free to deform, the Young’s modulus is 2.1 times higher
than for deformation of the whole system. This is comparable to the factor of ~1.5 found between
System I (III) with localised deformation and System II (IV) with global elongation obtained with PM6.
Conclusion
 Young’s modulus (E in TPa) and length (L in Å) of graphene using periodic calculations
(PBE-HSEh1PBE/6-31G*). For periodic calculations, different unit cells (a,b,c) gives almost identical results.
 Stress- Strain curve of the periodic , single & double layers, and supermolecular approach systems.
 Young’s modulus succesfully predicted with supermolecular and periodic approach.
 Strain-stress curves are similar to brittle materials, and almost identical (shape-wise) with the carbon fiber
stain-stress curves[4].
 Deformation on anchor atoms give higher Young’s moduli than elongating the whole sheet, as confirmed with
continuum mechanical simulations.
 Chemical bonding between graphene and the matrix will result in better mechanical properties in composites.
References
Acknowledgements
1.
2.
3.
4.
GVL acknowledges support as a Postdoctoral Fellow of the Research Foundation - Flanders (FWO), and from the COST Materials, Physical and
Nanosciences (MPNS) Action MP0901: “Designing Novel Materials for Nanodevices - from Theory to Practice (NanoTP)” (2009-2013 http://www.nanotp.org/).
Van Lier G.; Van Alsenoy C.; Van Doren V.; Geerlings P. Chem Phys Lett. 326, 181–5 (2000)
Batsanov S.S. Inorganic Materials. 37,871-885 (2001)
Shokrieh M. M. Materials and Design 31, 790–795 (2010)
Koizol, K.; Vilatela, J.; Moisala, A.; Motta, M.; Cunniff, P.; Sennett, M.; Windle, A.; Science 318, 1892-5 (2007)