Mechanical properties of graphene

This paper reviews the mechanical properties of graphene with particular attention to what is established and what is still uncertain. It clarifies the thickness and the elastic constants, and by also considering also phonon frequencies, it argues that “best values” come from graphite, when available. Properties not available from graphite include bending stiffness; this can be determined from studies of carbon nanotubes as well as graphene. In many ways, nanotubes provide access to fundamental properties of graphene, not least because they are the only form of graphene that can be unsupported (unstrained) in vacuum. Environmental effects are considered, including both interactions with substrates and with other solid and liquid media, which may affect the geometrical parameters defining graphene and associated elastic constants. Major uncertainties persist whether slipping or sticking dominates experimental observation, both between graphene and solid media, and between the layers of bilayer and multilayer graphene. The paper concludes with a short discussion of continuum and atomistic models of graphene.

the bending stiffness; this can be determined from studies of carbon nanotubes as 23 well as graphene. In many ways nanotubes provide access to fundamental properties 24 of graphene, not least as they are the only form of graphene that can be unsupported   to define the in-plane 2D elastic stiffness tensor c 2D ij = c ij d 0 with i, j = 1, 2, and d 0 is the 170 graphite interlayer spacing at ambient pressure. This tensor comes simply from the sp 2 bond 171 bending and stretching stiffnesses, and so is independent of the graphene thickness d per se.

172
To see this, consider making the graphene layers in graphite thinner, spaced more closely 173 and reducing a 33 , as happens under pressure. Then there are more graphene layers per 174 unit volume, a −1 33 , and the 3D constants c 11 and c 12 are increased proportionately -leaving 175 c 2D ij unchanged. The out-of-plane elastic constants, particularly c 33 but also c 13 , have to be 176 considered separately. This is done in Sec. III C.

177
A crucial aspect is then to know if the graphene sp 2 bonds -which largely determine 178 c 2D ij -are significantly influenced either by the environment of the graphene (what it is in moduli reported in Table II were obtained from inelastic X-ray scattering, 20 but similar 190 values were reported from ultrasonic and static mechanical testing. 21 191   TABLE II. Elastic moduli values obtained from inelastic X-ray scattering 20 in Voigt notation given there. Indeed, the in-plane Young's modulus Y 2D = 362 Nm −1 is consistent with 207 the Y 2D = 340 ± 50 Nm −1 of monolayer freestanding graphene measured by atomic force 208 microscopy (AFM) 25 , which will be discussed in Sec. IV A. 209 For graphene in vacuum, the greatest difference from graphite is likely to be an increase 210 in the thickness, as the π-orbitals are no longer compressed by the vdW attractive force -211 indeed, as discussed above, the thickness becomes defined by whatever convention is used 212 to specify where the π-orbitals end. The 3D elastic constants will vary inversely with the clearly robust. This is actually true of small molecules generally, which like graphene are 236 "all surface", yet whose vibrational frequencies are little affected from the vapour phase, 237 through solvation or liquefaction, to crystalline solid forms. It is not surprising, then, that 238 of-plane stiffness is calculated to be, not that of graphite, but only about half of it. 35 This 269 softness is attributed to the squeezing of π-orbitals through the graphene plane in a bilayer 270 system, whereas such squeezing-through is prohibited in graphite (infinite number of lay-271 ers) by symmetry. 35 In addition to the compliance of the Pauli exclusion for undeformed 272 π-orbitals, the other contribution from the graphene to the total compliance is from the 273 deformation of the π-orbitals of the graphene. This could be estimated by calculating the 274 energy difference between relaxed and deformed π-electron distributions. 275 In the absence of a conventionally-defined elastic constant c 33 based on internuclear dis-276 tances, one approach to define the out-of-plane stiffness of graphene is to use a related 277 quantity that is itself unambiguously defined and measurable. The in-plane bonds stiffen 278 under compressive in-plane strain, which can be expressed as a 2D strain and converted to 279 a 2D stress by c 2D ij . That has been measured by the increase in G-mode phonon frequency 280 under pressure. 31,34,[36][37][38] In graphene as in graphite, the 2D in-plane stress can be applied by 281 hydrostatic pressure, and the 2D stress is then directly proportional to the thickness. Since bending stiffness of graphene. However, the low bending stiffness is not necessary to explain 328 the quadratic dispersion relation. A transverse wave on a string (or sheet) with zero tension 329 but a bending stiffness D has ω(k) = k 2 (D/ρ) 1/2 , so the quadratic dispersion relation is fully for a 10 µm diameter graphene sheet.

339
The upper limit k max should be set so as to integrate over all possible phonon modes, in 340 reality the upper limit of the integral is set by the Bose-Einstein distribution term dropping 341 to zero upon increasing k, E.
Here, we write β = 1/k B T for convenience. The densities of states in 1D, 2D and 3D 343 systems, per unit area/volume, are given by: It has been shown that the value of the integral in Eq.

1D
2D 3D To our knowledge, the consequences of this divergence have not been studied experimen-369 tally in 1D systems. Certainly its observation would require the existence of an extremely 370 long system to ensure k min → 0, and for the system to be free-standing to allow these 371 phonons to propagate. The nearest humankind has got to a genuinely 1D system is carbyne The role of ripples in ensuring stability can be understood in terms of the restoring forces.

397
The softness of the ZA mode shown in Fig. 1, compared to the in-plane modes, is because of 398 the lack of restoring forces due to bond-stretching in the low amplitude limit, and of those 399 due to bending in the large-wavelength limit. The curvature induced by the ripples ensures 400 that there is some restoring force due to bond-stretching even in the low amplitude limit, 401 making the mode -partially -analogous to the radial breathing mode in SWCNTs. 55 is due solely to vdW forces. In any case, the potential will always have three key features in 412 common with the Lennard-Jones potential: (1) It will be attractive for moderate values of 413 r, with a minimum at r = r 0 , the inter-atomic separation in the absence of phonon effects.

415
As a result of these features, V (r) is not symmetric about r = r 0 and this asymmetry will, 416 in the absence of other effects, favour thermal expansion rather than contraction.

417
This argument applies directly to any reasonably isotropic and dense 3D solid and, for 418 that matter, a 2D solid existing in a 2D world (in which case out-of-plane phonon modes 419 would not exist). However, graphene's position is as that of a 2D solid in a 3D world. In 420 this case, the excitation of an out-of-plane vibration does not cause any thermal expansion in the out-of-plane direction. However it can cause contraction in the in-plane direction as 422 atoms are pulled inwards by the out-of-plane movement.

423
Thus, for graphene to exhibit a negative coefficient of thermal expansion coefficient 424 (CTE), all that is necessary is for the contribution from the out-of-plane phonons to dom-425 inate over that from the in-plane phonons. We can see how this can be the case at low 426 temperature from Fig. 1 where α(P, T ) is the pressure and temperature-dependent lattice parameter, as projected In 3D systems, while the elastic constants discussed above are properties of a material, a 494 bending stiffness is a property, not of a material, but of a structure, i.e., related to geometry.
there is a large literature in which an effective Young's modulus Y ef f and an effective thick-

501
ness h ef f are introduced such that both the in-plane elastic moduli and the bending stiffness 502 can be expressed: where in the second expression for D it is the plane-strain modulus that is used, as is correct 504 for a plate made of an isotropic material.

505
The model of Eq. 5 has had remarkable success in capturing the behaviour of graphene When a sheet of graphene is folded over onto itself, it adheres due to the vdW interaction.

539
The radius of the fold is determined by the strength of the vdW attraction and by the value of 540 D, and is of the order of the radius of C 60 . An example is found in large-diameter SWCNTs, 541 which are collapsed already at ambient pressure into the shape called "dogbone" or "peanut".
where D is the bending stiffness of the multilayer per unit length, d vdW is the equilibrium 557 distance between two graphene layers, Y is the Young's modulus, γ surface adhesion energy  Table I. It corresponds to a spring constant k = 0.419 N·m −1 for an unit cell.

579
The existence of this mode shows the corrugation of the graphene surface at the atomic

601
The applied force can then be calculated. However, the exact theory is far from simple 91 602 and many authors have used equations that appear to be over-simplified. The following 603 equation, for example, has been frequently used, 25 where R is the radius of the circular well, δ is the indentation depth, σ 2D 0 is the pre-tension the breaking strength can be calculated from: where R tip is the radius of the AFM tip and F max is the force at which the membrane breaks.
614 cular hole, is the so-called beam bending method, where the 2D membrane is now in the 616 form of a beam (or a stripe) and is suspended over a trough in the substrate. In this case 617 the load-deformation relationship is: 92 where w, t, and L are the width, thickness and length of the beam, σ 0 is the intrinsic stress, ing. Its intensity is usually very weak, and requires resonance condition to be observable.

698
Resonance Raman spectroscopy is particularly useful to study graphene and CNTs samples, 699 where the resonance condition is that the energy of the in-coming or out-going laser matches brane. This leads to a spherical blister with a radius R. AFM is used to measure the blister (compare Raman methods in Sec. IV B). The relation between its height, δ, and the pressure difference inside and out, ∆p, is, 113 where E is the Young's modulus, and d is the graphene thickness. K(ν) is a coefficient 737 depending on Poisson's ratio only and is very close to 3. Thus the elastic modulus can be 738 calculated from the measured AFM deflections.

739
The group of Bunch 114 first measured the adhesion of graphene on a silicon oxide substrate these configurations: an ensemble of graphene layers in suspension and considered separated 845 each from the others (Fig. 5(a.1)) or supporting an individual graphene layer on a hollowed 846 substrate ( Fig. 5(a.2)). In both cases, we may note that the graphene may not be flat due to 847 the spontaneous formation of ripples (Fig. 5(a.1)) and wrinkles (Fig. 5(a.2)). 127,128 Wrinkles 848 and ripples differ by their aspect ratio. 127 Ripples are isotropic, with an amplitude ∼ nm, 849 and an aspect ratio ∼ 1. Wrinkles are more aligned and larger, having an aspect ratio > 10, 850 due to the partial decoupling of bending and stretching modes. 50,127,128 851 In Fig. 5 (a.2), while the graphene sheet may be under tension at ambient pressure 852 (resulting in wrinkles with an axis perpendicular to the trench), 127 we might expect that  (1) see text (2) The sample were films with a mixture of monolayer, bilayer and few-layer graphene and having 2D characteristic signature of few-layer graphene rather than single-layer graphene.
(4) A gold microscopy grid was used to suspend graphene.
non-zero tension, 129 due to this, and analysed by Lu and Dunn.  All the studies in Table IV were  This is the case depicted in Fig. 5 (b) Graphene stiffness is one of the largest-ever measured, with a Young's modulus of ∼ 1 TPa 908 (Table II) Table I  role on the starting stress+doping state of graphene. In the following however, we will not 965 consider substrates in which this type of interaction may happen. Pressure-induced doping 966 effects will then be due essentially to interaction with the PTM and will be discussed in the following subsection. One should also note that the (in)commensurability of the lattices 968 should also play a role in the friction between graphene and its substrate.

969
(e) All these parameters may be impacted by the modification of the graphene-substrate 970 distance due to the application of pressure. However, in pressure ranges for which the PTM 971 is liquid (hydrostatic pressure), the graphene Raman response is always linear, 34,37,38,40,131,133 972 which tends to show that this effect is limited at pressures below ∼ 10 GPa.

973
The mechanical response of graphene to high pressures is usually followed through in situ after solidification it will be pressed down into the trench, as the PTM will typically be 1098 much more compressible than the substrate, and this will put the graphene under high 1099 tensile strain.  stacking. In graphite the vdW distance between graphene layers is 3.35Å which may be 1160 considered as the graphene thickness in that particular case (see Sec. II A and Table I). In 1161 bilayer graphene grown on a SiC(0001) surface, the measured graphene-graphene distance is The thickness of graphene in these different cases is to be related to the extension of its π-1168 orbitals. This is certainly a point of view in rupture with the Galilean continuum mechanics 1169 approach, but wholly consistent with the modern approach to the radius of atoms. In where the u 1 and u 2 are the relative displacement of the two carbon atoms along the two 1196 equivalent in-plane directions, as the hexagonal lattice of graphene is isotropic in-plane.

1197
When an additional graphene layer is added, Eq. 11 becomes 167 where u 3 and u 4 are the displacement of the two carbon atoms in the added layer, and C From the measured frequencies of E 1u and E 2g of graphite, we can calculate the ω 0 =1583.5 1207 cm −1 for the G-mode of a graphene plane in graphite -the G-mode frequency of graphene should be slightly higher than graphite, even their in-plane stiffnesses are the same.

1209
To quantify the effect of deformation of the π-orbitals on the G-mode frequency, we can 1210 introduce out-of-plane strain and calculate the shift of G-mode frequency. The off-diagonal 1211 term C in Eq. 12 for interlayer coupling can be expanded in terms of out-of-plane strain 1212 ε zz . The diagonal terms can be expanded too, to account for the possible modification of 1213 the in-plane sp 2 bond stiffness by the compression of the π-orbitals, and the solution to the secular equation is: where γ is the Grüuneisen parameter and SDP is the shear deformation potential. The i.e. ∼6.5 % greater than the vdW graphite distance. 1243 We may then conclude that different schemes of vdW stacking lead to changes in the is c 2D 11 . In contrast, the approach of Mahan, 173 modelling with a three-dimensional isotropic 1303 plate, invokes not only c 11 but also c 12 , which is incorrect. The potential energy per unit 1304 length of tube at the extreme of a sinusoidal motion r = A cos ωt is while the kinetic energy at the center of the motion is where N = 3.8 × 10 19 is the number of carbon atoms of mass m 0 in a unit area of graphene.

1307
Equating U max and E max , and rearranging, we have    for DWCNTs, one has to further assign Raman peaks to the outer or the inner tube, and 1424 the interaction between inner and outer tubes modifies the Kataura plot, 172 . In particular, 1425 Hirschmann et al. 189 showed that the wall-to-wall distance between inner and outer tubes in 1426 DWCNTs increases with increasing tube diameters, which makes the RBM upshifts of the 1427 inner tubes from intertube interaction no longer a constant, as most earlier work supposed.

1428
This requires further caution on the assignment of RBMs to inner tubes, but can be used 1429 to refine our calculations if needed.
1430 profile, shifting with pressure at different rates. It is tempting to assign these two components 1432 to outer and inner tubes for two reasons, one is that stress transmitted to the inner tube 1433 should be lower than hydrostatic pressure, resulting in two different responses to pressure, 1434 and the other is that outer (or inner) tubes in resonance at the same condition can have very 1435 close diameters and they should response similarly to pressure. In this section we have seen that much could be learned about the mechanical proper- AFM nanoindentation (see also Fig. 3 (a)).
Turning to quantum-mechanical models, the simple tight-binding description is used for obtaining the electronic structure 201 but is not suitable for mechanical properties.

1529
DFT provides models that can be made to replicate experimental data excellently. to be similar to graphite and those expected to be different from graphite -and anomalies.

1539
Graphene is commonly called a 2D material, which implies a thickness tending to zero.

1540
However, the π-electrons above and below the 2D plane of carbon nuclei extend the electron 1541 density of monolayer graphene into the third dimension, perpendicular to the 2D plane. For 1542 example, we can define a vdW thickness of graphene, 3.35Å, which is the experimentally 1543 measured spacing of graphene layers in graphite. One key conclusion is that, far from being 1544 a 2D material, graphene has a well-defined 3D structure, which may be modelled in various 1545 ways to help understand its mechanical properties. That is not to say that it cannot display 1546 2D physics, much as can a 100Å quantum well -which has a 3D physical structure of e.g.

1547
GaAs sandwiched between GaAlAs. Following from that, those of its mechanical properties  Due to the small sample size of exfoliated graphene -at least out-of-plane -experiments 1557 to measure many of its mechanical properties requires special design. In addition, the 1558 experimental data, from determining factors as substrates transferring strain to graphene, 1560 to subtle modification by influencing the π-orbital distribution.

1561
There are many derivative structures from graphene, in a way making the extraordinary 1562 properties of graphene tunable. They can also be used to help understand the properties 1563 of graphene. Among those, measurements on carbon nanotubes in some circumstances give 1564 the most accurate values for mechanical properties of graphene, perhaps even better than 1565 measurements on graphene itself, as nanotubes can be self-supporting, free-standing, and 1566 stable, thus excluding many of those complexities.