Elastic Properties and Stability of Physisorbed Graphene
Abstract
:1. Introduction
2. Teachings of Hooke’s Law and the Kirchhoff-Love Theory
3. Teachings of Phonon Dispersion Laws
4. Results of Calculations
5. Experimental Nanomechanical Measurements
6. Buckling Instability
7. A Buckling Mode of Non-Supported Graphene: Scrolling
8. Nonlocal, Nonlinear and Temperature Effects
9. Conclusions
Conflicts of Interest
Appendix: Force Constant of the Girifalco Interaction Potential
Substrate | h (nm) | a (meV nm4) | b (meV nm10) | z0 (nm) |
---|---|---|---|---|
gr(001) | 0.3354 | 4.14 | 6.37 × 103 | 0.338 |
h-BN(001) | 0.3331 | 3.25 | 5.49 × 103 | 0.343 |
Acknowledgments
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A (N/m) Equation (2) | ν Equation (2) | C (N/m) Equation (4) | g (N/m) Equation (3) | D (eV) Equation (7) | ν′ Equation (7) | Method |
---|---|---|---|---|---|---|
372 | 0.125 | 366 | 163 | Elastic constants of graphite [20]: A = hC11, = C12=C11, , g = h(C11 C22)=2 with h the graphite interlayer distance (0.335 nm) | ||
368 | 149 | 1.75 | HREELS phonon dispersion measurements [21] | |||
405 | 156 | 1.85 | Phonons dispersion (derived from experimental data) [22] | |||
363 | 146 | 1.33 | Phonons dispersion (GW-DFT calculations) [23] radial breathing mode of carbon SWNTs (see text) | |||
350 ± 65 | ||||||
0.412 | 236 | 83 | 0.85 | 1st generation Brenner potential [24,25] | ||
0.91 | 2nd generation Brenner potential [4] | |||||
289 | 0.397 | 243 | 1.39 | 2nd generation Brenner potential [16,25,26] | ||
355 | 0.15 | 1.1 | Monte-Carlo, empirical bond-order potential [3,27] | |||
0.22 | 320 | DFT-derived in-plane force field [28] | ||||
0.26 | 312 | 122 | AMBER 3D force field [29] | |||
410 | 1.88 | Non-orthogonal tight-binding [30] | ||||
0.24 | 358 | 1.10 | Non-orthogonal tight-binding [31] | |||
407 | 0.281 | Non-orthogonal tight-binding [32] | ||||
1.62 | Non self-consistent ab-initio [33] | |||||
377 | Hartree Fock 6-31G* [34] | |||||
0.149 | 345 | DFT Gaussian orbitals [35] | ||||
370 | 1.53–1.65 | LDA SIESTA [36] | ||||
1.61 | 0.56 | DFT tight-binding [37] | ||||
0.17 | 336 | DFT ABINIT code[38] | ||||
358 | 0.169 | DFT VASP program [39] | ||||
361 | 0.184 | 350 | LDA Vienna VASP [40] | |||
0.20 | 350 | DFT AIMPRO code [41] | ||||
95 | Measurement of torsional modulus [42] | |||||
330 ± 15 | Experimental AFM nano-indentation [43] | |||||
365 | 0.205 | 350 | 145 | 1.6 | Best fit (see Section 9) |
References | C11 | C12 | C13 | C33 | C44 | C66 |
---|---|---|---|---|---|---|
Reference [45] | 1060(20) | 180(20) | 15(5) | 36.5(10) | 0.18–0.35 | 440(40) |
Reference [20] | 1109(16) | 139(36) | 0(3) | 38.7(7) | 5.0(3) | 485(10) |
ct (km/s) | cl(km/s) | α(m2/s) | Phonon band structure |
---|---|---|---|
14.0 | 22.0 | 6.1 × 10−7 | HREELS measurements [21] |
14.3 | 23.1 | 6.25 × 10−7 | semi-experimental [22] |
13.8 | 21.8 | 5.28 × 10−7 | ab-initio calculations [23] |
Substrate | ZA phonon frequency at Γ (cm−1) [72] | K(J/m4) | (N/m) | (nm) | |
---|---|---|---|---|---|
Co(001) | 295 | 2.35 × 1021 | 49.1 | 0.64 | 0.14 |
Ni(111) | 240 | 1.56 × 1021 | 40.0 | 0.71 | 0.11 |
Cu(111) | 45 | 5.48 × 1019 | 7.5 | 1.64 | 0.021 |
SiO2 | [a] | 1.5 × 1020 | 12 | 1.3 | 0.036 |
gr(0001) | [b] | 1.08 × 1020 | 10.5 | 1.39 | 0.030 |
h-BN(0001) | [b] | 7.60 × 1019 | 8.8 | 1.51 | 0.025 |
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Lambin, P. Elastic Properties and Stability of Physisorbed Graphene. Appl. Sci. 2014, 4, 282-304. https://doi.org/10.3390/app4020282
Lambin P. Elastic Properties and Stability of Physisorbed Graphene. Applied Sciences. 2014; 4(2):282-304. https://doi.org/10.3390/app4020282
Chicago/Turabian StyleLambin, Philippe. 2014. "Elastic Properties and Stability of Physisorbed Graphene" Applied Sciences 4, no. 2: 282-304. https://doi.org/10.3390/app4020282
APA StyleLambin, P. (2014). Elastic Properties and Stability of Physisorbed Graphene. Applied Sciences, 4(2), 282-304. https://doi.org/10.3390/app4020282