Overview on the Evaluation of the Elastic Properties of Non-Carbon Nanotubes by Theoretical Approaches
Abstract
:1. Introduction
2. Atomic Structure of N-CNTs
- zigzag NTs (n, 0) when θ = 0° and m = 0;
- armchair NTs (n, n) when θ = 30° and n = m;
- chiral NTs (n, m) when 0° < θ < 30° and n ≠ m.
3. Analysis of the Literature Results
3.1. Elastic Constants of N-CNTs
3.1.1. Young’s and Shear Moduli
- at first, the Young’s modulus decreases and then becomes almost stable for > 1.5 nm [36];
3.1.2. Poisson’s Ratio
3.2. Vibtational Properties of N-CNTs
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Compound | BN | AlN | GaN | InN | BP | AlP | GaP | InP | SiC |
---|---|---|---|---|---|---|---|---|---|
aA1–A2, nm | 0.1447 [83] 0.145 [84] 0.147 [37] 0.151 [85] 0.153 [33] | 0.177 [37] 0.179 [84] 0.185 [40] 0.193 [86] 0.195 [87] | 0.175 [88] 0.184 [37] 0.185 [84] 0.186 [40] 0.194 [86] | 0.203 [1] 0.206 [84] | 0.183 [84] 0.193 [86] | 0.234 [89] 0.240 [37] | 0.220 [37] 0.225 [84] 0.229 [90] 0.236 [86] | 0.246 [84] 0.256 [86] | 0.177 [84] 0.179 [51] 0.185 [86] |
Approach | Year | Reference | Method | Type of NTs 1 | E, TPa 2 | Es, TPa⋅nm | G, TPa 2 | Gs, TPa⋅nm | Comment | |
---|---|---|---|---|---|---|---|---|---|---|
Atomistic | 1998 | Hernandez et al. [60] | TBMD | BN | (n, n) | 0.894 | – | – | – | average value |
(n, 0) | 0.896 | |||||||||
2003 | Moon et al. [51] | MD: Tersoff empirical potential | SiC | (n, n) | 0.621 | – | – | – | average value | |
(n, 0) | 0.558 | |||||||||
2004 | Jeng et al. [38] | MD: TB many body potential | GaN | (5, 5) | 0.793 | – | – | – | – | |
(9, 0) | 0.721 | |||||||||
2004 | Kang and Hwang [40] | MD: Tersoff-type potential | BN | (5, 5) | 0.870 | – | – | – | – | |
AlN | 0.453 | |||||||||
GaN | 0.796 | |||||||||
2007 | Baumeier et al. [46] | ab initio: DFT-SIC | BN | (n, n) | – | 0.278 | – | – | converged average value | |
(n, 0) | 0.272 | |||||||||
SiC | (n, n) | 0.167 | ||||||||
(n, 0) | 0.162 | |||||||||
2007 | Verma et al. [55] | MD: TB potential | BN | (n, n) | 1.107 | – | 0.965 | – | average value | |
(n, 0) | 1.044 | 1.555 | ||||||||
2009 | Santosh et al. [59] | MD: force—constant approach | BN | (n, n); (n, 0) | 1.017 | – | 0.326 | – | converged average value | |
2009 | Setoodeh et al. [52] | MD: Tersoff potential | SiC | (n, n) | – | 0.182 | – | average value | ||
(n, 0) | 0.180 | |||||||||
2009 | Pan and Si [53] | MD: Tersoff bond order potential | SiC | single crystalline | 0.465 | – | – | – | tn = 0.30 nm | |
0.540 | tn = 0.90 nm | |||||||||
2010 | Zhou et al. [54] | SiC | 0.641 | – | – | – | tn = 0.89 nm | |||
0.595 | tn = 1.69 nm | |||||||||
0.582 | tn = 2.49 nm | |||||||||
2011 | Zhang et al. [61] | MD: DFTB | BN | (n, n) | 0.840 | – | 0.366 | – | converged average value | |
(n, 0) | 0.844 | 0.368 | ||||||||
2014 | Le [43] | MD: harmonic force fields | BN | (n, n) | – | 0.282 | – | – | converged average value | |
(n, 0) | 0.281 | |||||||||
SiC | (n, n) | 0.148 | ||||||||
(n, 0) | 0.145 | |||||||||
2015 | Hao et al. [47] | ab initio: LCAO | AlN | (n, n) | 0.360 | – | – | – | converged average value | |
(n, 0) | 0.340 | |||||||||
2015 | Xiong and Tian [50] | MD, Tersoff potential: force approach | BN | (n, n) | – | – | – | 0.315 | average value | |
(n, 0) | 0.329 | |||||||||
energy approach | (n, n) | 0.281 | ||||||||
(n, 0) | 0.292 | |||||||||
2015 | Tao et al. [56] | MD: TB potential + FE model | BN | (n, n) | 0.911 | – | – | – | converged average value | |
(n, 0) | 0.930 | |||||||||
2017 | Kochaev [37] | ab initio | BN | (n, n) | – | 0.347 | – | – | average value | |
(n, 0) | 0.340 | |||||||||
AlN | (n, n) | 0.253 | ||||||||
(n, 0) | 0.247 | |||||||||
GaN | (n, n) | 0.207 | ||||||||
(n, 0) | 0.193 | |||||||||
AlP | (n, n) | 0.172 | ||||||||
(n, 0) | 0.159 | |||||||||
GaP | (n, n) | 0.131 | ||||||||
(n, 0) | 0.106 | |||||||||
2019 | Pinhal et al. [29] | DFT + B3LYP | AlN | (20, 20) | 0.393 | – | – | – | – | |
(20, 0) | 0.387 | |||||||||
(20, 10) | 0.392 | |||||||||
GaN | (20, 20) | 0.383 | ||||||||
(20, 0) | 0.367 | |||||||||
(20, 10) | 0.370 | |||||||||
2020 | Choyal et al. [31] | MD: TB potential | BN | (10, 10) | 1.053 | – | – | – | Ln ≈ 21 nm | |
(17, 0) | 1.066 | |||||||||
2020 | Vijayaraghavan and Zhang [35] | MD: REBO | BN | (10, 10) | 2.8 | – | – | – | tn = 0.105 nm | |
CM | 2010 | Oh [62] | CL thermodynamic approach + TB potential | BN | (n, n) | 0.960 | – | – | – | converged average value |
(n, 0) | 0.975 | |||||||||
NCM/MSM | 2006 | Li and Chou [34] | beams + FE model | BN | (n, n) | 0.916 | – | 0.465 | – | converged average value |
(n, 0) | 0.913 | 0.475 | ||||||||
2011 | Jiang and Guo [33] | “stick-and-spring” model + closed-form solution | BN | (n, n) | 0.270 | 0.095 | converged average value | |||
(n, 0) | 0.262 | 0.088 | ||||||||
2015 | Ansari et al. [66] | analytical solution | BN | (n, n) | 0.825 | – | – | – | average value | |
(n, 0) | 0.823 | |||||||||
2015 | Yan and Liew [70] | representative cell | BN | (n, n) | 0.970 | – | 0.416 | – | converged average value | |
(n, 0) | 0.967 | 0.418 | ||||||||
2016 | Giannopoulos et al. [69] | springs + FE model: free vibrations | BN | (12, 12) | 0.592 | – | – | – | Ln ≈ 11 nm | |
(21, 0) | 0.523 | |||||||||
2016 | Jiang and Guo [39] | “stick-and-spring” model + analytical | BN | (n, n) | – | 0.278 | – | – | converged average value | |
(n, 0) | 0.276 | |||||||||
AlN | (n, n) | 0.121 | ||||||||
(n, 0) | 0.120 | |||||||||
GaN | (n, n) | 0.120 | ||||||||
(n, 0) | 0.119 | |||||||||
BP | (n, n) | 0.118 | ||||||||
(n, 0) | 0.117 | |||||||||
GaP | (n, n) | 0.060 | ||||||||
(n, 0) | 0.059 | |||||||||
InP | (n, n) | 0.051 | ||||||||
(n, 0) | 0.051 | |||||||||
SiC | (n, n) | 0.169 | ||||||||
(n, 0) | 0.168 | |||||||||
2018 | Salavati et al. [65] | beams + FE model | BN | (n, n); (n, 0) | 0.928 | – | – | – | converged average value | |
2019 | Yan et al. [71] | longitudinal and torsional vibrations | BN | (n, n); (n, 0); (n, m) | 0.972 | 0.418 | – | converged average value | ||
2019 | Genoese et al. [41] | “stick-and-spring” model + Donnell thin shell model | BN | (n, n) | – | 0.255 | – | 0.092 | converged average value | |
(n, 0) | 0.250 | 0.104 | ||||||||
SiC | (n, n) | 0.152 | 0.053 | |||||||
(n, 0) | 0.149 | 0.061 | ||||||||
2021 | Sakharova et al. [36] | beams + FE model | BN | (n, n); (n, 0); (n, m) | 0.984 | – | 0.486 | – | converged average value | |
2022 | Zakaria [68] | two-section beams + FE model | BN | (12, 12) | 0.538 | 0.108 | Ln ≈ 11 nm | |||
(21, 0) | 0.489 | 0.117 | ||||||||
Experimental | 1998 | Chopra and Zettl [75] | TEM: thermal vibrational amplitude | MWBNNTs | 1.220 ± 0.240 | – | – | – | – | |
2004 | Suryavanshi et al. [76] | TEM: electric-field-induced resonance | MWBNNTs with = 34–94 nm | 0.722 ± 0.140 | – | – | – | average value for 18 MWBNNTs | ||
2005 | Hung et al. [16] | NS + analytical | SWGaNNTs | 0.484 | – | – | – | Ln = 500 nm | ||
0.223 | Ln = 300 nm | |||||||||
2007 | Goldberg et al. [77] | AFM-TEM: bending + analytical | MWBNNTs with = 40–100 nm | 0.5–0.6 | – | – | – | average value | ||
2009 | Stan et al. [15] | CR-AFM + FEA | faceted AlNNTs with triangular cross-section | 0.3252 ± 0.015 | – | – | – | inner facet | ||
0.3050 ± 0.013 | outer facet | |||||||||
2010 | Ghassemi et al. [78] | AFM-TEM: bending + analytical | MWBNNTs with = 38–51 nm | 0.5 ± 0.1 | – | – | – | average value for 5 NTs | ||
2011 | Arenal et al. [74] | HRTEM-AFM + analytical | SWBNNTs | 1.11±0.17 | – | – | – | tn = 0.07 nm | ||
0.87±0.13 | tn = 0.09 nm | |||||||||
0.25±0.04 | tn = 0.34 nm | |||||||||
2013 | Tanur et al. [79] | AFM: a three-point bending + analytical | MWBNNTs with = 18–55 nm | 0.760 ± 0.03 | – | 0.07 ± 0.01 | – | E in bending, average value (0.1 ± 0.02 to 1.8 ± 0.3 TPa) for 20 NTs | ||
2019 | Zhou et al. [80] | HRTEM: high-order resonance | MWBNNTs with = 28–57 nm | 0.906 | – | – | – | average value | ||
2019 | Chen et al. [81] | TEM: force transducer holder + analytical | MWBNNT with = 37.34 nm and 40 layers | 1.050–1.370 | – | – | – | E calculated from tree compression cycles |
Approach | Year | Reference | Method | Type of NTs 1 | ν | Comment | |
---|---|---|---|---|---|---|---|
Atomistic | 1998 | Hernandez et al. [60] | TBMD | BN | (n, n) | 0.260 | average value |
(n, 0) | 0.240 | ||||||
2004 | Jeng et al. [38] | MD: TB many body potential | GaN | (5, 5) | 0.263 | – | |
(9, 0) | 0.221 | ||||||
2007 | Verma et al. [55] | MD: TB potential | BN | (n, n), (n, 0) | 0.140 | average value | |
2017 | Kochaev [37] | ab initio | BN | (10, 10) | 0.560 | – | |
(10, 0) | 0.570 | ||||||
AlN | (10, 10) | 0.520 | |||||
(10, 0) | 0.550 | ||||||
GaN | (10, 10) | 0.530 | |||||
(10, 0) | 0.550 | ||||||
AlP | (10, 10) | 0.510 | |||||
(10, 0) | 0.510 | ||||||
GaP | (10, 10) | 0.510 | |||||
(10, 0) | 0.520 | ||||||
CM | 2010 | Oh [62] | CL thermodynamic approach + TB potential | BN | (n, n) | 0.150 | converged average value |
(n, 0) | 0.170 | ||||||
NCM/MSM | 2015 | Ansari et al. [66] | analytical solution | BN | (n, n), (n, 0) | 0.217 | average value |
2016 | Jiang and Guo [39] | “stick-and-spring” model + analytical | BN | (n, n) | 0.216 | converged average value | |
(n, 0) | 0.219 | ||||||
AlN | (n, n) | 0.281 | |||||
(n, 0) | 0.287 | ||||||
GaN | (n, n) | 0.285 | |||||
(n, 0) | 0.290 | ||||||
BP | (n, n) | 0.360 | |||||
(n, 0) | 0.365 | ||||||
GaP | (n, n) | 0.428 | |||||
(n, 0) | 0.435 | ||||||
InP | (n, n) | 0.455 | |||||
(n, 0) | 0.460 | ||||||
SiC | (n, n) | 0.095 | |||||
(n, 0) | 0.100 | ||||||
2019 | Genoese et al. [41] | “stick-and-spring” model + Donnell thin shell model | BN | (n, n) | 0.239 | converged average value | |
(n, 0) | 0.226 | ||||||
SiC | (n, n) | 0.330 | |||||
(n, 0) | 0.331 | ||||||
2021 | Sakharova et al. [36] | beams + FE model | BN | (n, n); (n, 0); (n, m) | 0.150 | converged average value | |
Experimental | 2005 | Hung et al. [16] | NS + analytical | SWGaNNTs | 0.242 | – |
Approach | Year | Reference | Method | Type of NTs | Support Case | Ln, nm | fn1, THz | Comments | |
---|---|---|---|---|---|---|---|---|---|
Atomistic | 2015 | Ansari and Ajori [57] | MD: TB potential | BN | (10, 10) | CF | 6 | 0.145 | The BNNTs with clamped-clamped support have higher values of than those with cantilevered support. The first fundamental frequency decreases for small nanotube length and then tends to stabilize for NT length Ln > 4 nm. |
8 | 0.081 | ||||||||
10 | 0.048 | ||||||||
12 | 0.016 | ||||||||
CC | 6 | 0.855 | |||||||
8 | 0.532 | ||||||||
10 | 0.274 | ||||||||
12 | 0.210 | ||||||||
2015 | Chandra et al. [48] | MD: Tersoff-type potential | BN | (10, 10) | CC | 7 | 0.52 | The values of fn1 were determined at T = 400K. The bigger the BNNT length, the higher the first fundamental frequency. | |
21 | 0.06 | ||||||||
CM | 2013 | Panchal et al. [63] | thin wall tube (outer diameter of 0.8 nm, thickness of 0.065 nm) + analytical | BN | – | CF | 6 | 0.744 | The values of fn1 were obtained for the case of attached mass at free NT end of 10−8 fg (from the range of 10−8 to 10−2 fg). The fn1 value increases with decreasing of the attached mass and NT length. |
8 | 0.419 | ||||||||
10 | 0.268 | ||||||||
NCM/MSM | 2006 | Li and Chou [34] | beams + FE model | BN | (4, 4) | CF | 7 | 0.044 | The first fundamental frequency values decrease with increased in the nanotube length and diameter. The decrease rate is higher in the case of clamped-clamped support. |
9 | 0.030 | ||||||||
11 | 0.019 | ||||||||
13 | 0.015 | ||||||||
CC | 7 | 0.289 | |||||||
9 | 0.178 | ||||||||
11 | 0.122 | ||||||||
13 | 0.089 | ||||||||
(7, 0) | CF | 7 | 0.044 | ||||||
9 | 0.030 | ||||||||
11 | 0.019 | ||||||||
13 | 0.015 | ||||||||
CC | 7 | 0.281 | |||||||
9 | 0.174 | ||||||||
11 | 0.119 | ||||||||
13 | 0.085 | ||||||||
2010 | Chowdhury et al. [98] | unspecified elastic elements | BN | (4, 4) | CC | 6 | 1.647 | The fn1 frequency decreases with increasing NT length, Ln, and diameter, Dn. | |
8 | 1.253 | ||||||||
9 | 1.118 | ||||||||
10 | 0.941 | ||||||||
(8, 0) | 6 | 1.794 | |||||||
8 | 1.382 | ||||||||
10 | 1.029 | ||||||||
12 | 0.824 | ||||||||
2013 | Panchal et al. [67] | beams + FE model | BN | (5, 5) | CF | 6 | 0.077 | The values of fn1 de-crease with increasing Ln and Dn The fundamental frequencies for high-er-order vibrational modes were also cal-culated. | |
9 | 0.037 | ||||||||
12 | 0.021 | ||||||||
13 | 0.016 | ||||||||
CC | 6 | 0.426 | |||||||
9 | 0.227 | ||||||||
12 | 0.135 | ||||||||
13 | 0.107 | ||||||||
2014 | Khani et al. [100] | beams + FE model | SiC | (15, 15) | CF | 4 | 0.025 | The fn1 frequency decreases with increasing SiCNT length.First five vibrational modes shapes were analysed and respective values of the fundamental frequencies were calculated. The fundamental frequencies are higher for the case of clamped-clamped support. | |
6 | 0.021 | ||||||||
8 | 0.011 | ||||||||
10 | 0.008 | ||||||||
CC | 4 | 0.062 | |||||||
6 | 0.044 | ||||||||
8 | 0.029 | ||||||||
10 | 0.025 | ||||||||
(13, 0) | CF | 4 | 0.024 | ||||||
6 | 0.013 | ||||||||
8 | 0.006 | ||||||||
10 | 0.004 | ||||||||
CC | 4 | 0.060 | |||||||
6 | 0.039 | ||||||||
8 | 0.023 | ||||||||
10 | 0.015 | ||||||||
2016 | Giannopoulos et al. [69] | springs + FE model | BN | (12, 12) | CF | 8 | 0.210 | The values of fn1 decrease with increasing NT length, Ln, and diameter, Dn. The fundamental frequencies for first three modes were calculated. The fn values decrease with increasing of the vibrational mode order. The fundamental frequencies are higher for the case of clamped-clamped support. | |
16 | 0.124 | ||||||||
25 | 0.073 | ||||||||
35 | 0.061 | ||||||||
CC | 8 | 0.408 | |||||||
16 | 0.256 | ||||||||
25 | 0.171 | ||||||||
35 | 0.123 | ||||||||
(20, 0) | CF | 8 | 0.207 | ||||||
16 | 0.125 | ||||||||
24 | 0.081 | ||||||||
32 | 0.064 | ||||||||
CC | 8 | 0.467 | |||||||
16 | 0.256 | ||||||||
24 | 0.174 | ||||||||
32 | 0.127 | ||||||||
2017 | Ansari and Rouhi [101] | beams + FE model | SiC | (5, 5) | CF | 4 | 0.359 | The first fundamental frequency, fn1 decreases for small values of the NT lengths and then tends to stabilize for Ln > 3 nm. The values of fn1 obtained for clamped-clamped support are about two times higher than those obtained for clamped-free support. | |
6 | 0.236 | ||||||||
8 | 0.154 | ||||||||
9 | 0.103 | ||||||||
CC | 4 | 0.780 | |||||||
6 | 0.565 | ||||||||
8 | 0.458 | ||||||||
9 | 0.390 | ||||||||
(9, 0) | CF | 4 | 0.462 | ||||||
6 | 0.205 | ||||||||
8 | 0.134 | ||||||||
9 | 0.103 | ||||||||
CC | 4 | 0.878 | |||||||
6 | 0.565 | ||||||||
8 | 0.390 | ||||||||
9 | 0.390 | ||||||||
2019 | Yan et al. [71] | analytical solution + Euler beam theory | BN | (5, 5) | CF | 5 | 1.052 | The values of fn1 decrease with increasing Ln. fn1 slightly increases for small NT diameters and then tends to nearly constant value for Dn > 1.0 nm. | |
6 | 0.918 | ||||||||
7 | 0.731 | ||||||||
CC | 5 | 2.127 | |||||||
6 | 1.860 | ||||||||
7 | 1.487 | ||||||||
2022 | Zakaria [68] | two-section beams + FE model | BN | (12, 12) | SS | 8 | 0.189 | The values of fn1 decrease with increasing NT length, Dn, and diameter, Dn. | |
16 | 0.054 | ||||||||
25 | 0.022 | ||||||||
35 | 0.012 | ||||||||
(20, 0) | 8 | 0.179 | |||||||
16 | 0.048 | ||||||||
24 | 0.023 | ||||||||
32 | 0.012 |
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Antunes, J.M.; Pereira, A.F.G.; Sakharova, N.A. Overview on the Evaluation of the Elastic Properties of Non-Carbon Nanotubes by Theoretical Approaches. Materials 2022, 15, 3325. https://doi.org/10.3390/ma15093325
Antunes JM, Pereira AFG, Sakharova NA. Overview on the Evaluation of the Elastic Properties of Non-Carbon Nanotubes by Theoretical Approaches. Materials. 2022; 15(9):3325. https://doi.org/10.3390/ma15093325
Chicago/Turabian StyleAntunes, Jorge M., André F. G. Pereira, and Nataliya A. Sakharova. 2022. "Overview on the Evaluation of the Elastic Properties of Non-Carbon Nanotubes by Theoretical Approaches" Materials 15, no. 9: 3325. https://doi.org/10.3390/ma15093325