# Critical Output Torque of a GHz CNT-Based Rotation Transmission System Via Axial Interface Friction at Low Temperature

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{M}) can drive the inner tube (rotor) to rotate in the outer tubes. When the axial gap between the motor and the rotor was fixed, the friction between their neighbor edges was stronger at a lower temperature. Especially at temperatures below 100 K, the friction-induced driving torque increases with ω

_{M}. When the rotor was subjected to an external resistant torque moment (M

_{r}), it could not rotate opposite to the motor even if it deformed heavily. Combining molecular dynamics simulations with the bi-sectioning algorithm, the critical value of M

_{r}was obtained. Under the critical torque moment, the rotor stopped rotating. Accordingly, a transmission nanosystem can be designed to provide a strong torque moment via interface friction at low temperature.

## 1. Introduction

## 2. Results and Discussion

#### 2.1. Rotation Transmission of the Zigzag Model

_{Tran}, as shown in Figure 1, of the zigzag nanosystem with M

_{r}= 0 eV at different conditions. In this case, the resistant torque moment was only from the two stators. The rotor has no axial translation because the axial translation freedom of its right edge was fixed. The results are listed in Table 1. When the temperature of the system was higher than 100 K, R

_{Tran}decreased with increasing ω

_{M}. For example, at 500 K, it decreased from ~0.96 at ω

_{M}= 50 GHz to ~0.11 at ω

_{M}= 200 GHz. Hence, both ω

_{M}and temperature influenced the rotational frequency of the rotor. The reason is that the interaction at the interface edges depends on the two main factors. For example, at high temperature, the tubes have thermal expansion both in the axial and the radial directions. Considering both axial thermal expansion (thermal expansion coefficient is negative when T < 300 K, otherwise positive when T > 300 K [39]) and the existence of edge barriers [40], the distance between the interface edges of the motor and the rotor was less than 0.5 nm. When the distance was very close to 0.34 nm, their interaction became very low according to the 12-6 type Lennard-Jones potential in the AIREBO potential for describing the nonbonding interaction between atoms. When the distance is between 0.2 nm and 0.34 nm, strong repulsion will increase the interface friction, and further leads to larger rotational frequency of the rotor, i.e., ω

_{R}. However, as mentioned above, the friction between the rotor and the stators increased with ω

_{R}. Therefore, the lower value of ω

_{M}led to a higher value of R

_{Tran}when the temperature was not lower than 100 K according to the results listed in Table 1.

_{Tran}increased with the input rotational frequency on the motor when the temperature was lower than 100 K. For example, at 50 K, it increased from ~0.54 at ω

_{M}= 50 GHz to ~0.80 at ω

_{M}= 200 GHz. The reason is that, at a lower temperature, thermal vibration of the edge atoms introduced weaker axial collision between the motor and the rotor. Meanwhile, the friction from the two stators did not increase with ω as high as that at a higher temperature, i.e., M

_{M}(ω) − (M

_{S1}+ M

_{S2}) grew with the increasing of ω. Hence, R

_{Tran}increased with ω at 50 K or a lower temperature. Briefly, as the temperature decreased, the temperature effect on the inter-tube friction between the rotor and the stators became weaker, and the rotation transmission efficiency was mainly determined by the rotor’s rotation when the right edge of the rotor was confined in its axial movement.

_{r}

^{cr}of the zigzag system at different conditions with the bi-section algorithm and listed the results in Table 2. It states that the torque moment transferred via the interface was lower at a higher temperature when ω

_{M}was constant. In fact, two special phenomena attracted our attention. One was the sudden jump of the critical value of the applied resistant torque moment for a system with specified ω

_{M}in a cooling/heating process. For instance, when ω

_{M}= 150 GHz, the critical value was ~175 eV at 50 K. If the temperature increased to 100 K or higher, the critical value became less than 1.0 eV. Similarly, when ω

_{M}= 200 GHz, the critical value varied from ~130 eV at 100 K to ~0.5 eV at 300 K. The reason is that the motor provided strong friction with the motor at their end interface at low temperature because the friction from the stators can be neglected at such low temperature and very low rotational speed of rotor according to Equation (10). It implies that a strong power can be output by the rotor via the interface friction with the high speed rotating motor at low temperature. It also suggests an easy way to control the output just changing the temperature of the system.

_{M}was higher than 100 GHz and the resistant torque moment was too large (Figure 2c). It means that the transmission efficiency became higher at a lower temperature for the same system. From the snapshots shown in Figure 2, we can find that the rotor tube did not deform obviously when M

_{r}= −175 eV at 50 K. The two tubes rotated oppositely (Supplementary Material Movie 1). The motor rotated around within 7 ps but the rotor needed about 3700 ps for a round of rotation, which is much lower than that of the motor. However, at 8 K, if M

_{r}= −205 eV, the rotor deformed obviously after about 50 ps of rotation. Its loading end was buckled and enclosed at 56 ps. Later the whole tube was buckled (Supplementary Material Movie 2). It was not caused by the friction from the stators. This can be verified by the state of the system at 8 K with ω

_{M}= 200 GHz, i.e., the value of R

_{Tran}>0.5 when M

_{r}= -217.13 eV. It means that the rotor has over 100 GHz of the rotational frequency with the same direction as that of the motor when 217.13 eV of resistant torque was applied on its free end. If the resistant torque moment increased slightly, e.g., from 217.13 eV to 217.23 eV, the rotor tube buckled rapidly.

#### 2.2. Rotation Transmission of the Armchair Model

_{M}was larger than 100 GHz, the value of R

_{Tran}decreased with the increasing temperature (Table 3). Hence, the temperature effect on the armchair model was identical to that on the zigzag model. Compared with the zigzag model, the armchair model had better rotation transmission efficiency, but poor stability of the rotor’s rotation at 500 K or when ω

_{M}was lower than 100 GHz. This is because the potential barriers on the armchair surface were aligned with generatrix. At high temperature, thermal vibration increased the friction from the stator but decreased the friction from the motor via their end interface. Hence, rotation transmission was not stable, which could be verified from the fluctuation of the curves in Figure 3.

_{M}= 50 GHz. At 50 K, the critical value was ~0.75 eV when ω

_{M}= 100 GHz. According to the results listed in Table 4, the critical value decreased with the increasing temperature when the system had a fixed value of ω

_{M}. For instance, when ω

_{M}= 50 GHz, M

_{r}

^{cr}is ~ 0.35 eV at 50 K and became 0.075 eV at 500 K.

_{M}= 150 GHz, M

_{r}

^{cr}is 1.0 eV at 100 K and jumped up to 130 eV at 50 K. for the same system, the rotor buckled when M

_{r}= 260 eV before reversing its rotational direction. At temperature below 100 K, the rotor could not be stopped by the external resistant torque moment when the motor had a high rotational frequency, e.g., 150 GHz or higher. Hence, cooling the system will lead to stronger torque transferred from the motor to the rotor via their end interface. This conclusion is identical to that for the zigzag model, as well.

## 3. Model and Methods

#### 3.1. Model

_{M}, also influences the transmission effect. Hence, 5 temperatures between 8 K and 500 K together with 4 input rotation between 50 GHz and 200 GHz were considered in the tests.

_{r}was very small. Second, if M

_{r}was too large, the rotor had an opposite rotational direction to the motor. Finally, when M

_{r}was in an interval, the rotor’s rotational frequency, i.e., ω

_{R}, was far less than ω

_{M}. Hence, we could find the lower boundary of the interval using the bi-section algorithm. This value is called the critical resistant torque moment and labeled as “M

_{r}

^{cr}”. The major task of this study is to measure the quantity of M

_{r}

^{cr}on the rotor under different conditions.

#### 3.2. Methodology

#### 3.2.1. Mechanism of Rotation Transmission

_{M}”. When the rotor is rotating with the motor along the same direction, friction-induced resistant moments from the two stators are increasing. If there is no active moment applied on the right edge of the rotor, the rotational frequency of the rotor can be expressed as

_{z}is the moment of inertia of the rotor about Z-axis. When the driving moment is balanced by the resistant moment, the rotor has a stable rotational frequency (SRF), i.e.,

_{S1}and M

_{S2}have the same direction as M

_{M}, and the rotational frequency of the rotor is negative, i.e.,

_{S1}and M

_{S2}will be much less than M

_{M}or M

_{r}when the rotor has no rotation on time average. And we conclude that the M

_{M}is approximately equal to M

_{r}when the absolute value of ω

_{R}is much larger than ω

_{M}. According to this conclusion, we build a model to test the transmission torque via the interface between the motor and the rotor.

#### 3.2.2. Bi-Section Algorithm for Finding M_{r}^{cr}

_{r}

^{cr}can be found when the rotor’s rotational frequency is far less than that of the motor. For simplicity, the rotation transmission ratio (RTR) of the system is defined as

_{r}

^{cr}reads

_{r}

^{cr}, the bi-section algorithm is adopted. In this algorithm, an initial interval of M

_{r}, e.g., [a

_{0}, b

_{0}], should be provided. When considering anticlockwise rotation implies a positive value of ω

_{R}and ω

_{M}, a

_{0}and b

_{0}should satisfy the following inequation,

_{i}

_{+1}, i.e.,

_{i}, b

_{i}] satisfies Equation (7);

_{i+}

_{1}using Equation (8), find the value of ω

_{R}(c

_{i+}

_{1});

_{i+}

_{1}as the value of M

_{r}

^{cr}.

#### 3.2.3. Molecular Dynamics Simulation Approach

#### 3.2.4. Temperature Effect on Torque Transmission

#### 3.2.5. Effect of Input Rotation on Torque Transmission

_{R}. Hence, the motor can also provide stronger friction on the rotor by the input of higher rotational frequency. This is another way to change the output torque moment via the interface for the same system.

_{r}= 0, on the rotor, which was in stable rotation, the resistant torque from the 2 stators could balance the active torque moment from the motor. Consider both torque moments as functions in terms of the rotational frequencies, their relationship yields

_{M}is twice of ω

_{R}, it means that the resultant resistant torque moment from the two stators is equal to that from the motor via the interface.

_{r}≠ 0, and ω

_{R}= 0, according to Equations (3) and (4) we have M

_{S1}(ω

_{R}) = M

_{S2}(ω

_{R}) = 0, and

_{M}is a dual quantity of ω

_{M}with respect to the output power of the motor via the interface.

## 4. Conclusions

_{M}) and a nano-bearing from double-walled CNTs. In the nano-bearing, the outer tubes were partly fixed as stators, and the inner tube acted as a rotor. Without applying external resistant torque moment (M

_{r}) on it, the rotor would be driven to rotate with the motor via friction at their end interface. Using molecular dynamics simulations together with the bi-sectioning algorithm, the critical value of M

_{r}was obtained. Under the critical torque moment, the rotor’s rotational frequency was zero or much less than ω

_{M}. When searching the critical values, two important phenomena were discovered.

_{r}has a sudden jump when varying temperatures. It increases more than 100 times when temperature decreased from 100 K to 50 K or to 8 K. This implies that one can adjust the output torque from the rotor by decreasing the system temperature. The other is that the rotor cannot rotate opposite to the motor even if it was buckled by strong external resistant torque moment. More importantly, the phenomena are independent on the chirality of the CNTs in the system.

## Supplementary Materials

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

- Aust, R.B.; Drickamer, H.G. Carbon: A new crystalline phase. Science
**1963**, 140, 817–819. [Google Scholar] [CrossRef] - Iijima, S. Helical microtubules of graphitic carbon. Nature
**1991**, 354, 56–58. [Google Scholar] [CrossRef] - Bonard, J.M.; JeanPaul, S.; Tomas, S. Why are carbon nanotubes such excellent field emitters. Ultramicroscopy
**1998**, 73, 7–10. [Google Scholar] [CrossRef] - Cumings, J.; Zettl, A. Low-friction nanoscale linear bearing realized from multiwall carbon nanotubes. Science
**2000**, 289, 602–604. [Google Scholar] [CrossRef] - Fennimore, A.M.; Yuzvinsky, T.D.; Han, W.Q.; Fuhrer, M.S.; Cumings, J.; Zettl, A. Rotational actuators based on carbon nanotubes. Nature
**2003**, 424, 408–410. [Google Scholar] [CrossRef] - Feng, J.; Li, W.B.; Qian, X.F.; Qi, J.S.; Qi, L.; Li, J. Patterning of graphene. Nanoscale
**2012**, 4, 4883–4899. [Google Scholar] [CrossRef] - Zhang, R.F.; Ning, Z.Y.; Zhang, Y.Y.; Zheng, Q.S.; Chen, Q.; Xie, H.H.; Zhang, Q.; Qian, W.Z.; Wei, F. Superlubricity in centimetres-long double-walled carbon nanotubes under ambient conditions. Nat. Nanotechnol.
**2013**, 8, 912–916. [Google Scholar] [CrossRef] - Kroto, H.W.; Heath, J.R.; O’Brien, S.C.; Curl, R.F.; Smalley, R.E. C60 buckminsterfullerene. Nature
**1985**, 318, 162–163. [Google Scholar] [CrossRef] - Novoselov, K.S.; Geim, A.K.; Morozov, S.V.; Jiang, D.; Zhang, Y.; Dubonos, S.V.; Grigorieva, I.V.; Firsov, A.A. Electric field effect in atomically thin carbon films. Science
**2004**, 306, 666–669. [Google Scholar] [CrossRef] - Jiang, J.W.; Park, H.S. Negative poisson’s ratio in single-layer black phosphorus. Nat. Commun.
**2014**, 5, 4727. [Google Scholar] [CrossRef] - Cai, K.; Wan, J.; Wei, N.; Cai, H.F.; Qin, Q.H. Thermal stability of a free nanotube from single-layer black phosphorus. Nanotechnology
**2015**, 27, 235703. [Google Scholar] [CrossRef] - Chen, N.; Lusk, M.T.; van Duin, A.C.T.; Goddard, W.A. Mechanical properties of connected carbon nanorings via molecular dynamics simulation. Phys. Rev. B
**2005**, 72, 085416. [Google Scholar] [CrossRef] [Green Version] - Guo, W.L.; Zhong, W.Y.; Dai, Y.T.; Li, S.N. Coupled defect-size effects on interlayer friction in multiwalled carbon nanotubes. Phys. Rev. B
**2005**, 72, 075409. [Google Scholar] [CrossRef] - Bichoutskaia, E.; Heggie, M.I.; Popov, A.M.; Lozovik, Y.E. Interwall interaction and elastic properties of carbon nanotubes. Phys. Rev. B
**2006**, 73, 045435. [Google Scholar] [CrossRef] - Lee, C.; Wei, X.D.; Kysar, J.W.; Hone, J. Measurement of the elastic properties and intrinsic strength of monolayer graphene. Science
**2008**, 321, 385–388. [Google Scholar] [CrossRef] - Zhang, H.W.; Cai, K.; Wang, L. Deformation of single-walled carbon nanotubes under large axial strains. Mater. Lett.
**2008**, 62, 3940–3943. [Google Scholar] [CrossRef] - Ozden, S.; Autreto, P.A.S.; Tiwary, C.S.; Khatiwada, S.; Machado, L.; Galvao, D.S.; Vajtai, R.; Barrera, E.V.; Ajayan, P.M. Unzipping carbon nanotubes at high impact. Nano Lett.
**2014**, 14, 4131–4137. [Google Scholar] [CrossRef] - Chernozatonskii, L.A.; Sorokin, P.B.; Kuzubov, A.A.; Sorokin, B.P.; Kvashnin, A.G.; Kvashnin, D.G.; Avramov, P.V.; Yakobson, B.I. The influence of size effect on the electronic and elastic properties of diamond films with nanometer thickness. J. Phys. Chem. C
**2011**, 115, 132–136. [Google Scholar] [CrossRef] - Kvashnin, A.G.; Chernozatonskii, L.A.; Yakobson, B.I.; Sorokin, P.B. Phase diagram of quasi-two-dimensional carbon, from graphene to diamond. Nano Lett.
**2014**, 14, 676–681. [Google Scholar] [CrossRef] - Zou, J.; Ji, B.; Feng, X.Q.; Gao, H. Self-assembly of single-walled carbon nanotubes into multiwalled carbon nanotubes in water: Molecular dynamics simulations. Nano Lett.
**2006**, 6, 430–434. [Google Scholar] [CrossRef] - Cook, E.H.; Buehler, M.J.; Spakovszky, Z.S. Mechanism of friction in rotating carbon nanotube bearings. J. Mech. Phys. Solids
**2013**, 61, 652–673. [Google Scholar] [CrossRef] - Zheng, Q.S.; Jiang, Q. Multiwalled carbon nanotubes as gigahertz oscillators. Phys. Rev. Lett.
**2002**, 88, 045503. [Google Scholar] [CrossRef] - Legoas, S.B.; Coluci, V.R.; Braga, S.F.; Coura, P.Z.; Dantas, S.O.; Galvao, D.S. Molecular-dynamics simulations of carbon nanotubes as gigahertz oscillators. Phys. Rev. Lett.
**2003**, 90, 055504. [Google Scholar] [CrossRef] - Rivera, J.L.; Mccabe, C.; Cummings, P.T. Oscillatory behavior of double-walled nanotubes under extension: A simple nanoscale damped spring. Nano Lett.
**2003**, 3, 1001–1005. [Google Scholar] [CrossRef] - Legoas, S.B.; Coluci, V.R.; Braga, S.F.; Coura, P.Z.; Dantas, S.O.; Galvão, D.S. Gigahertz nanomechanical oscillators based on carbon nanotubes. Nanotechnology
**2004**, 15, S184–S189. [Google Scholar] [CrossRef] - Rivera, J.L.; McCabe, C.; Cummings, P.T. The oscillatory damped behaviour of incommensurate double-walled carbon nanotubes. Nanotechnology
**2005**, 16, 186–198. [Google Scholar] [CrossRef] - Motevalli, B.; Liu, J.Z. Tuning the oscillation of nested carbon nanotubes by insertion of an additional inner tube. J. Appl. Phys.
**2013**, 114, 56–58. [Google Scholar] [CrossRef] - Cai, K.; Yin, H.; Qin, Q.H.; Li, Y. Self-excited oscillation of rotating double-walled carbon nanotubes. Nano Lett.
**2014**, 14, 2558–2562. [Google Scholar] [CrossRef] - Barreiro, A.; Rurali, R.; Hernández, E.R.; Moser, J.; Pichler, T.; Forró, L.; Bachtold, A. Subnanometer motion of cargoes driven by thermal gradients along carbon nanotubes. Science
**2008**, 320, 775–778. [Google Scholar] [CrossRef] - Zambrano, H.A.; Walther, J.H.; Jaffe, R.L. Thermally driven molecular linear motors: A molecular dynamics study. J. Chem. Phys.
**2009**, 131, 241104. [Google Scholar] [CrossRef] [Green Version] - Santamaría-Holek, I.; Reguera, D.; Rubi, J.M. Carbon-nanotube-based motor driven by a thermal gradient. J. Phys. Chem. C
**2013**, 117, 3109–3113. [Google Scholar] [CrossRef] - Zhang, S.; Liu, W.K.; Ruoff, R.S. Atomistic simulations of double-walled carbon nanotubes (DWCNTs) as rotational bearings. Nano Lett.
**2010**, 4, 293–297. [Google Scholar] [CrossRef] - Rueckes, T.; Kim, K.; Joselevich, E.; Tseng, G.Y.; Cheung, C.L.; Lieber, C.M. Carbon nanotube-based nonvolatile random access memory for molecular computing. Science
**2000**, 289, 94–97. [Google Scholar] [CrossRef] - Peng, H.B.; Chang, C.W.; Aloni, S.; Yuzvinsky, T.D.; Zettl, A. Ultrahigh frequency nanotube resonators. Phys. Rev. Lett.
**2006**, 97, 087203. [Google Scholar] [CrossRef] - Garcia-Sanchez, D.; Paulo, A.S.; Esplandiu, M.J.; Perez-Murano, F.; Forró, L.; Aguasca, A.; Bachtold, A. Mechanical detection of carbon nanotube resonator vibrations. Phys. Rev. Lett.
**2007**, 99, 085501. [Google Scholar] [CrossRef] - Lassagne, B.; Tarakanov, Y.; Kinaret, J.; Garcia-Sanchez, D.; Bachtold, A. Coupling mechanics to charge transport in carbon nanotube mechanical resonators. Science
**2009**, 325, 1107–1110. [Google Scholar] [CrossRef] - Cai, K.; Yin, H.; Wei, N.; Chen, Z.; Shi, J. A stable high-speed rotational transmission system based on nanotubes. Appl. Phys. Lett.
**2015**, 106, 021909. [Google Scholar] [CrossRef] - Shi, J.; Cai, K.; Qin, Q.H. A nanoengine governor based on the end interfacial effect. Nanotechnology
**2016**, 27, 495704. [Google Scholar] [CrossRef] - Jiang, H.; Liu, B.; Huang, Y.; Hwang, K.C. Thermal expansion of single wall carbon nanotubes. J. Eng. Mater. Technol.
**2004**, 126, 265–270. [Google Scholar] [CrossRef] - Guo, Z.; Chang, T.; Guo, X.; Gao, H. Thermal-induced edge barriers and forces in interlayer interaction of concentric carbon nanotubes. Phys. Rev. Lett.
**2011**, 107, 105502. [Google Scholar] [CrossRef] - Cai, K.; Cai, H.F.; Shi, J.; Qin, Q.H. A nano universal joint made from curved double-walled carbon nanotubes. Appl. Phys. Lett.
**2015**, 106, 241907. [Google Scholar] [CrossRef] [Green Version] - Yin, H.; Cai, K.; Wei, N.; Qin, Q.H.; Shi, J. Study on the dynamics responses of a transmission system made from carbon nanotubes. J. Appl. Phys.
**2015**, 117, 234305. [Google Scholar] [CrossRef] [Green Version] - Plimpton, S. Fast parallel algorithms for short-range molecular dynamics. J. Compu. Phys.
**1995**, 117, 1–19. [Google Scholar] [CrossRef] - Large-Scale Atomic/Molecular Massively Parallel Simulator, LAMMPS. Available online: https://lammps.sandia.gov/ (accessed on 27 March 2019).
- Stuart, S.J.; Tutein, A.B.; Harrison, J.A. A reactive potential for hydrocarbons with intermolecular interactions. J. Chem. Phys.
**2000**, 112, 6472–6486. [Google Scholar] [CrossRef] [Green Version] - Nosé, S. A unified formulation of the constant temperature molecular dynamics methods. J. Chem. Phys.
**1984**, 81, 511–519. [Google Scholar] [CrossRef] [Green Version] - Hoover, W.G. Canonical dynamics: Equilibrium phase-space distributions. Phys. Rev. A Gen. Phys.
**1985**, 31, 1695–1697. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Historical curves of the rotation transmission ratio, i.e., R

_{Tran}in Equation (5), of the zigzag model with Gap = 1.5 nm and subjected to M

_{r}at different conditions. Note that the curves in each column are obtained at the same value of input rotation, e.g., “ω

_{M}= 50 GHz” means that the curves in the left column are corresponding to the system with input rotational frequency of 50 GHz. Each row contains the curves obtained at the same temperature, e.g., “T = 8 K” says the curves in the top row are obtained at 8 K. Not all the results involved in the bi-section algorithm were shown here. Only three of them in each case were listed for showing the critical values.

**Figure 2.**Relative rotation between motor and rotor in the zigzag model. (

**a**) The perspective of the initial configuration after relaxation. (

**b**) Snapshots of the system with M

_{r}= 175 eV, ω

_{M}= 150 GHz at 50 K. The rotor rotates a round after about 3700 ps. (

**c**) Snapshots of the system with M

_{r}= 205 eV, ω

_{M}= 100 GHz at 8 K. The rotor has obvious buckling at 50 ps, the loading end of the rotor was enclosed after 56 ps.

**Figure 3.**Historical curves of the rotation transmission ratio, i.e., R

_{Tran}, of the armchair model at different conditions. Not all the results involved in the bi-section algorithm were shown here. Only three of them in each case were listed for showing the critical values.

**Figure 4.**The initial model for testing the transmission torque moment from motor to rotor. (

**a**) Configuration of the transmission model. The left short carbon nanotube acts as a motor with constant input rotational frequency of ω

_{M}. The right part is a nano-bearing from double walled carbon nanotubes (CNTs). The rotor is confined by the two same concentric stators. The rotational frequency of the rotor, i.e., ω

_{R}, is the output rotation. At the right edge of the rotor, an active resistant torque moment, M

_{r}, is applied for reducing the value of ω

_{R}. All the tubes are concentric with the same axis of Z. The essential geometrical factors are as Gap = 1.5 nm being the distance between the motor and the left stator. L

_{S}is the length of a stator. L is the distance between the two stators. Axial commensurate CNTs are used to form an armchair model from armchair tubes (with C-C bonds as edges) or a zigzag model from zigzag tubes (with uniformly distributed atoms as edges). The neighboring edges of motor and rotor with an initial distance of d = 0.5 nm are hydrogenated. Detailed parameters are listed in Table 5. (

**b**) Free body diagram of the rotor. M

_{M}is the driving moment from the motor. M

_{S1}and M

_{S2}are from the two stators.

Temperature | ω_{M} = 50 GHz | ω_{M} = 100 GHz | ω_{M} = 150 GHz | ω_{M} = 200 GHz |
---|---|---|---|---|

T = 8 K | 0.60 ± 0.028 | 0.74 ± 0.013 | 0.82 ± 0.009 | 1.0 ± 0.025 |

T = 50 K | 0.54 ± 0.026 | 0.69 ± 0.048 | 0.80 ± 0.061 | 0.80 ± 0.061 |

T = 100 K | 0.91 ± 0.180 | 0.68 ± 0.057 | 0.79 ± 0.084 | 0.64 ± 0.045 |

T = 300 K | 0.95 ± 0.157 | 0.72 ± 0.082 | 0.77 ± 0.113 | 0.64 ± 0.081 |

T = 500 K | 0.96 ± 0.150 | 0.70 ± 0.202 | 0.76 ± 0.122 | 0.11 ± 0.030 |

**Table 2.**Critical values of the resistant moment, i.e., M

_{r}

^{cr}, on the rotor in the zigzag model. Unit of the moment: eV. Sign convention: Minus sign means the direction of M

_{r}is opposite to that of ω

_{M}.

Temperature | ω_{M} = 50 GHz | ω_{M} = 100 GHz | ω_{M} = 150 GHz | ω_{M} = 200 GHz |
---|---|---|---|---|

T = 8 K | −3.0 | −205(buckled) | −210(buckled) | −217.23(buckled) |

T = 50 K | −0.393 | −1.75 | −175 | −200(buckled) |

T = 100 K | −0.32 | −0.688 | −0.625 | −130 |

T = 300 K | −0.181 | −0.338 | −0.219 | −0.5 |

T = 500K | −0.188 | −0.25 | −0.188 | −0.3 |

**Table 3.**Statistical results, i.e., Mean ± Std, of R

_{Tran}of the armchair model and subjected to M

_{r}= 0 eV (see black lines in Figure 3).

Temperature | ω_{M} = 50 GHz | ω_{M} = 100 GHz | ω_{M} = 150 GHz | ω_{M} = 200 GHz |
---|---|---|---|---|

T = 8 K | 0.97 ± 0.124 | 0.99 ± 0.049 | 0.89 ± 0.016 | 0.93 ± 0.009 |

T = 50 K | 0.99 ± 0.089 | 0.71 ± 0.051 | 0.88 ± 0.071 | 0.67 ± 0.022 |

T = 100 K | 0.99 ± 0.091 | 0.71 ± 0.056 | 0.81 ± 0.167 | 0.67 ± 0.043 |

T = 300 K | 0.95 ± 0.138 | 0.68 ± 0.109 | 0.48 ± 0.051 | 0.37 ± 0.023 |

T = 500 K | 0.53 ± 0.097 | 0.52 ± 0.095 | 0.40 ± 0.050 | 0.36 ± 0.033 |

Temperature | ω_{M} = 50 GHz | ω_{M} = 100 GHz | ω_{M} = 150 GHz | ω_{M} = 200 GHz |
---|---|---|---|---|

T = 8 K | −3.0 | −250(buckled) | −260(buckled) | −275(buckled) |

T = 50 K | −0.35 | −0.75 | −130 | −250(buckled) |

T = 100 K | −0.28 | −0.469 | −1.0 | −100 |

T = 300 K | −0.125 | −0.19 | −0.25 | −0.5 |

T = 500 K | −0.075 | −0.175 | −0.15 | −0.25 |

**Table 5.**Parameters of the rotation transmission nanosystem with different models shown in Figure 4. Dimension unit: nm.

Model | L | Motor/Rotor | Stator/Stator | ||||||
---|---|---|---|---|---|---|---|---|---|

Chirality | Length | Diameter | Number of Atoms | Chirality | L_{S} | Diameter | Number of Atoms | ||

Zigzag | 4.75 | (26,0) | 2.20/9.87 | 2.04 | 572C+26H/2444C+26H | (35,0) | 1.56 | 2.74 | 560C/560C |

Armchair | 4.64 | (15,15) | 1.97/9.84 | 2.03 | 510C+30H/2430C+30H | (20,20) | 1.60 | 2.71 | 560C/560C |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wu, P.; Shi, J.; Wang, J.; Shen, J.; Cai, K.
Critical Output Torque of a GHz CNT-Based Rotation Transmission System Via Axial Interface Friction at Low Temperature. *Int. J. Mol. Sci.* **2019**, *20*, 3851.
https://doi.org/10.3390/ijms20163851

**AMA Style**

Wu P, Shi J, Wang J, Shen J, Cai K.
Critical Output Torque of a GHz CNT-Based Rotation Transmission System Via Axial Interface Friction at Low Temperature. *International Journal of Molecular Sciences*. 2019; 20(16):3851.
https://doi.org/10.3390/ijms20163851

**Chicago/Turabian Style**

Wu, Puwei, Jiao Shi, Jinbao Wang, Jianhu Shen, and Kun Cai.
2019. "Critical Output Torque of a GHz CNT-Based Rotation Transmission System Via Axial Interface Friction at Low Temperature" *International Journal of Molecular Sciences* 20, no. 16: 3851.
https://doi.org/10.3390/ijms20163851