A three-stage rotary transmission nanobearing driven by a gigahertz nanomotor

Carbon nanotubes¹ are a one-dimensional carbon material with a tubular shape and not only are of interest for basic scientific investigations, but also have the potential to completely change nanodevices, such as nano-transmission systems,^2,3 nano-oscillators,^4–7 nanometers, nano-bearings,^8–11 and nanomotors.^12–19 Carbon nanotubes and graphene¹ behave similarly in carbon allotropes.^1,20–24 First, they have extremely high modulus and strength.^25,26 Second, carbon nanotubes are super-lubricated.^27–29 In addition, interactions between carbon nanotubes are known. For example, the inter-shell friction is extremely low between two adjacent concentric carbon nanotubes. Finally, carbon nanotubes have a chiral character characterized by a helix angle.⁵ Chirality can be described by two chiral indices (n, m) and gives carbon nanotubes excellent mechanical properties, which make them attractive candidates for nanodevices.

In the nanomotor model based on carbon nanotubes, the rotor can be rotated by using different methods.^18,30–32 In 2015, Cai et al.^2,33 proposed the concept of a carbon nanotube rotary transmission system, in which short carbon nanotubes were fixed, and the longer internal carbon nanotube rotation frequency could reach the gigahertz range. In their model, a rotating carbon nanotube motor is laid coaxially with a nanobearing in a rotating transmission nanosystem. The nanomotor forces the rotor to rotate via van der Waals forces exerted on the adjacent edges. Their results show that the rotor in the bearing has axial freedom and can oscillate during the rotation. The output rotation frequency of the rotor in the nano-axis fluctuates significantly. To reduce the fluctuation range of the rotor rotation frequency and obtain a down-conversion transmission system, Qiu et al.³⁴ proposed in 2018 a two-stage rotating transmission system for carbon nanotubes. In their study, the system model consisted of two nanobearings arranged coaxially with a motor. Their results demonstrate that, when the radial difference between the two rotors is greater than 0.18 nm and less than 0.34 nm, the second rotor has a rotational transmission ratio between 0.1 and 0.9.

In the present study, we introduce a three-stage rotational transmission system for carbon nanotubes. The movement of the ends of rotor 1 and rotor 2 is constrained by more components, and the oscillation of rotors 1 and 2 is suppressed, as shown in Fig. 1. Unlike the models from Refs. 2 and 34, this model has three coaxial multiwalled carbon nanotubes; namely, 1, 2, and 3 bearings. To show the feasibility of rotational transmission and the rotational transmission efficiency, we consider herein two other factors: The first factor is the application of nanotube chirality to rotating components. In addition, the difference in radii is considered. The parameters are detailed in Table I. The second factor is the layout of the nanotubes in the three components. A detailed schematic diagram is shown in Fig. 1. See Section III for a discussion of the response of the system dynamics.

Figure 1 shows a schematic diagram of a three-stage rotary drive system with components made of carbon nanotubes. Table I lists the parameters of the carbon nanotubes involved in the model and the stator corresponding to the rotor in the nanobearing. A total of 17 models were built and tested.

Since the outer edge of each stator and the adjacent edges of the rotating member are hydrogenated, each carbon atom is covalently bonded to a hydrogen atom. Therefore, the edge carbon atoms are saturated and do not bond with other hydrogen atoms under normal conditions. The interaction between atoms and adjacent components in the system is a bondless interaction and is described by the Lennard–Jones potential.³⁵ The cutoff of the Lennard–Jones potential is 1.02 nm. σ_C−C=0.34 nm, ε_C−C=2.8 meV, ε_H−H=1.5 meV,

The interaction between carbon and hydrogen atoms is given by the AIREBO potential.³⁶ Figure 1 shows that the movement of the tube edge is limited to 1.0 nm. The difference in radius between the rotor and the stator in each multiwalled carbon nanotube is less than 0.34 nm, so the rotor does not escape from the stator because it is restrained by the edge of the stator.³⁷ The direction of the rotor axial force can be varied during the simulation, but this has no effect on the results.

According to the action and reaction law, in LAMMPS,³⁸ each carbon nanotube is defined as a set of atoms; that is, each atom is assigned to a single block. The output rotational frequencies of the rotors are given by

where $LRj$ is the angular-momentum vector of the rotors, $IRj$ is the moment of inertia tensor for the rotors, and $ωRj$ is the angular velocity of the rotors, with j = 1, 2, 3. The calculation includes all effects due to the passage of atoms through the boundaries.

The quantities $IRj$ and $LRj$ are given by

where m, r, v, and n are the atomic mass, vertical distance between the atom and the axis of rotation, atomic velocity, and the total number of atoms in a rotating part, respectively.

To show the efficiency of the rotary transmission of three rotors in a system, we define the rotational transmission ratios (RTRs)³⁹ as

where R_j is the RTR of rotor j, with j = 1, 2, 3.

The centroid of the carbon nanotubes is given by

where Z, m_i, and z_i are the axial center-of-mass coordinates, the atomic mass, and the atomic z coordinate of the carbon nanotube, respectively. The quantity n is the total number of atoms, and s is a measure of time.

The efficiency of the rotary transmission is categorized into three levels:³⁴ 

• R₃<0.1 or R₃>0.9 is transmission failure.

After 20 ns of rotation, if the rotational transmission ratio of rotor 3 is less than 0.1 or greater than 0.9, the deceleration of the transmission fails.

• (b)

  0.1<R₃<0.4 or 0.6<R₃<0.9 is acceptable for transmission.

After 20 ns of rotation, if the rotational transmission ratio of rotor 3 is less than 0.4 and greater than 0.1 or less than 0.9 and greater than 0.6, the transmission is acceptable. The system can provide down-converted transmission.

• (c)

  0.4<R₃<0.6 is excellent transmission.

After 20 ns of rotation, if R₃ is between 0.4 and 0.6, then the system is an ideal model of down-converted rotational transmission.

In the simulation by the open-source molecular dynamics package LAMMPS,³⁸ the rotational transmission efficiency of the system in the model is given. The time step of the atomic integration of Newton’s second law in the system is 0.001 ps. The simulation uses three steps to find the result: First, build the model (see Fig. 1), whose parameters are detailed in Table I, to minimize the energy of the system model and re-adjust the system configuration. Second, after minimizing the energy, the rotating part of the motor, rotors 1–3, have fixed carbon atoms at the left end, the fixed length is 0.5 nm, and all stators (carbon atoms) are fixed. The simulation of the Nosé–Hoover⁴⁰ thermal bath (200 ps time step) was continuously carried out in the standard NVT ensemble. Finally, after undoing the constraints on the rotating parts and the thermal bath, the left side of the motor is fixed at 0.5 nm axially and rotated at a frequency of 200 GHz for 20 ns by using a Nosé–Hoover thermostat to maintain the system temperature at 300 K.

In the rotational drive in models M1–M9, both the motor and rotors 1 and 2 have the same chirality, which differs from that of rotor 3. To better show the rotational transmission effect of the three rotors, we list in Fig. 2 the time-varying images of the RTR curves of rotors 1–3 for the nine models.

In models M1–M3, as shown in the first row of Fig. 2, the motor and rotors 1–3 are made of the same armchair-shaped carbon nanotubes, and the transmission efficiency of the system is gradually reduced as the diameter of the pipe increases. When the radius of the rotating part is less than 0.475 nm [i.e., the radius of the (7,7) carbon nanotubes in M1], the motor rotates synchronously with rotors 1–3 (RTR = 1.0) because they are concentrically arranged and the adjacent hydrogenated ends are intense. The van der Waals interaction is greater than that between the stator and the rotor. When the radius of the rotating component is between 0.543 nm (M2) and 0.881 nm (M3), the rotational speed of the motor differs significantly from that of the synchronously rotating rotors 1–3. The rotation ratio of the three rotors is between 0.49 and 0.75.

There are two reasons for this result: First, as the diameter of the pipe increases, the number of carbon atoms within a given width increases, and the resistance of the stator to the rotor increases. In other words, the driving torque is provided by the interaction between the atom at the right end of the motor and the atom on the left side of rotor 1. Second, because the left end of the motor is fixed, rotors 1–3 are only constrained by adjacent components and can move less than 0.4 nm in an axial direction. The ends of the three components are subjected to strong van der Waals forces, so rotors 1–3 rotate at the same speed. Based on the final rotational ratios of the three rotors in the three models, we conclude that the same armchair carbon nanotubes as the motor and rotors 1–3, with a radius between 0.543 nm and 0.881 nm, can provide a stable output that is different from the input of the motor.

For M4–M6, as shown in the second row of Fig. 2, the motor and rotors 1–3 are all made of identical zigzag carbon nanotubes. As the diameter of the nanotubes increases, the transmission efficiency of the system gradually decreases. When the radius of the rotating member is greater than 0.587 nm (M6), the rotational speed of the motor differs significantly from that of rotors 1–3, and the rotational transmission ratio is less than 0.75. Based on the final values of the rotational ratios of the three rotors in the three models, we conclude that the rotational transmission ratio is less than 0.75 when the radius is greater than 0.587 nm (M6). The radius of the rotating component is then adjusted to provide the motor in the transmission system. The input has a different stable output.

The armchair carbon nanotube stator provides higher friction than the sawtooth stator. For example, in M7–M9, the rotor ratio curve (see Fig. 2) shows that the motor and rotors 1 and 2 in M7 are made of the same armchair carbon nanotubes (8, 8), and rotor 3 is made of an armchair carbon nanotube (5, 5). The final rotation ratio of the rotor differs significantly from that of the motor. The motor and rotors 1 and 2 in M9 are made of the same zigzag carbon nanotubes (14, 0), rotor 3 is made of zigzag-shaped carbon nanotubes (8, 0), and the final rotors transmission ratio does not differ significantly from the motor. In other words, the rotors rotate synchronously with the motor. Therefore, the three-stage rotary transmission system can be designed by using armchair carbon nanotubes as the motor and rotors 1 and 2 and a zigzag carbon nanotube as rotor 3.

In models M8 and M9, the motor and rotors 1–3 are each made of zigzag carbon nanotubes (8, 0) and (14, 0). In M8, the larger-radius nanotube (14,0) is used for the motor and for rotors 1 and 2 and the smaller-radius nanotube (8,0) is used for rotor 3. The rotor ratio curve for M9 (see Fig. 2), shows that the motor and rotors rotate synchronously.

In M8, the transmission ratios of the motor and of rotors 1–3 differ slightly, and the transmission ratios of the three rotors are all 0.86. Therefore, we conclude that, when the motor uses the same smaller-radius serrated carbon nanotubes as rotors 1 and 2, and rotor 3 uses a larger-radius zigzag carbon nanotube, an available three-stage rotary transmission system can be obtained.

To explore the interaction between different chiral carbon nanotubes, a successful three-stage rotary transmission system with reduced rotational frequency was obtained. The simulation uses the same chirality for the carbon nanotubes of the motor and rotors 2 and 3, and this chirality differs from that of rotor 1. The rotational transmission ratio curves of the rotors are shown in Fig. 3 under the simulation conditions.

In Fig. 2, we know that, in the transmission system made of carbon nanotubes with the same chirality for rotors 1–3, if the radius of the carbon nanotubes is between 0.3 and 0.55, the transmission ratios of the motor and rotors 1–3 are about 1.0. Unlike the previous model, in Fig. 3, the motor and rotors 2 and 3 for M10–M14 are made of nanotubes with the same chirality, and the difference in radius with respect to rotor 1 is greater than 0.2 nm. Based on their transmission ratio curves for M10 and M11, we note two phenomena: First, the transmission ratios of the motor and the rotor differ significantly, and the rotation speed of the rotor is less than that of the motor, but rotor 2 is made of an identical carbon nanotube as the motor. Rotor 3 always rotates synchronously because, when the radial difference between the motor and rotor 2 is larger than 0.2 nm, the van der Waals interaction between the motor with rotor 2 is weakened. Second, the transmission ratio (0.75) of rotor 3 for M10 is greater than the transmission ratio (0.5) of rotor 3 for M11 because the carbon nanotubes (5, 5) in M11 are restrained by the armchair stator, which causes greater friction. Therefore, we conclude that, when the transmission system is made of rotor 1 and the other three rotating parts are made of carbon nanotubes of the same armchair configuration with different radii, the difference between their radii is greater than 0.2 nm, and a successful deceleration transmission system can be obtained.

For models M12 and M13, both the rotating part and the stator are made of zigzag carbon nanotubes. When a (14,0) carbon nanotube with large radius is used in M12 as rotor 1, the final gear ratios for the motor and rotors 1–3 do not differ significantly; that is, the motor and rotors rotate synchronously. However, when a small-radius (8,0) carbon nanotube is used in M13 as rotor 2, the transmission ratio of the motor is higher than rotors 1–3 (the transmission ratios of rotors 1–3, respectively, are 0.57, 0.48, and 0.46). The reason is that the difference in radius between the motor and rotor 1 in M13 and the difference in radius between rotor 1 and rotor 2 are both greater than 0.2, the interaction between them is weak, and rotor 2 has the same interaction with the radius of rotor 3. Therefore, rotor 3 is difficult to drive by rotor 2 (8, 0) with a smaller radius. This means that, when rotor 1 and the other three rotating members are both made of zigzag carbon nanotubes, and their radii are larger than 0.2 nm, an excellent reduction transmission system can be obtained.

In models M1–M3 and M4–M6, we know that, when the same carbon nanotubes are used as rotating parts and the tube diameter is larger than 0.6 nm, a usable reduction transmission system can be obtained. Carbon nanotubes with different chiral characteristics in models M10 and M11 also have an ideal rotor output response. Therefore, we choose a drive system with the same chirality but different radii for the carbon nanotubes of the motor and rotors 1–3 and establish four different models (M14–M17). The transmission ratio curves in Fig. 4 indicate the rotational transmission efficiency of the rotors.

In models M14 and M15, all rotating parts in the system are made of carbon nanotubes of different radii. When the radii of the motor to rotor 3 is sequentially increased (e.g., M14), the rotational speed of the three rotors is much smaller than that of the motor. The transmission ratio for rotor 3 is only 0.09 (i.e., less than 0.1), which means that the rotation transmission fails. The reason for this is that the radius of the armchair carbon nanotubes has increased, enhancing the friction between the stator and the rotor, and the small motor exerts only a small attractive force on the large-sized rotor, making it difficult to drive the rotor. When the radius of the motor to rotor 3 decreases (e.g., M15), the rotational speed of the rotor differs significantly from that of the motor, and the transmission ratios of rotors 1–3 have significant discrimination; their final values are 0.81, 0.51, and 0.29, respectively. Thus, transmission systems are viable for two reasons: First, the motor in M15 interacts strongly with rotor 1, which is stronger than the force exerted in M14 by the motor on rotor 1. As shown in Fig. 5, the COM coordinate in M15 is obviously smaller than that in M14. It is smaller than the centroid of the rotor in M14. Second, the difference between the adjacent rotating parts in M14 is between 1.5 and 2.0 nm, and the interaction between them is weakened. Therefore, we conclude that, for carbon nanotubes with armchair chiral features, the ratio of the radius of the input end to that of the output end is reduced, and the difference in radii between the rotating parts is greater than 0.2 nm, which allows us to obtain a usable transmission system.

When the rotating parts are made of zigzag carbon nanotubes of different radii (see M16 and M17 of Fig. 4), the transmission ratio curve of the rotor shows that, when the radii of the rotating parts differ by 0.15 nm, regardless of the motor or the rotor, the radius of rotor 3 changes, resulting in excellent down-conversion output. The transmission ratios of rotor 3 in M16 and M17 are 0.45 and 0.60, respectively. The deceleration effect in M16 is more obvious than that in M17 because the rotor of M16 is attracted by the adjacent rotor and stator ends, as shown for M16 in Fig. 6 (Multimedia view). The axial movement gradually approaches the motor, and the interaction between adjacent ends is enhanced, in M16. The middle rotor 1 does not easily drive a large rotating part through a small-sized motor, and although the rotor is attracted in M17, the motor gradually approaches it along the z axis, as in M17 in Fig. 6 (Multimedia view), but it is driven by a motor of a larger size. Thus, rotor 3 eventually attains a large rotational speed, so we conclude that the rotating parts are made of zigzag carbon nanotubes of differing radii. An excellent deceleration transmission system can thus be obtained when the difference in radii between adjacent rotating parts is greater than 0.15 nm.

In a three-stage rotary drive system, a 200 GHz motor is powered at a constant temperature of 300 K. Considering the chirality and radial differences of the rotating component carbon nanotubes, a rotating transmission system model consisting of 17 different carbon nanotubes was tested to obtain an effective down-conversion transmission rotation system. Based on the rotor rotation ratios for each model, we arrive at the following conclusions for this nanostructure design:

• If the rotating parts are made of identical carbon nanotubes, the rotor output rotation is determined by the radius of the rotating parts. When the radius is between 0.58 and 0.88 nm, the rotors have a rotational transmission ratio (RTR) between 0.1 and 0.9.

• When the radial difference between a rotor and other rotating parts is greater than 0.2 nm, the design is usable.

• Because of the greater friction between armchair carbon nanotubes, the armchair rotor reduces the output rotation frequency more effectively than does a zigzag rotor.

• A larger rotating part can easily drive a smaller rotating part, but a smaller rotating part is difficult to drive a larger rotating part. In this case, an effective down-converting rotary drive system can be obtained by combining rotating parts with different radii.

This work was supported by the National Science Foundation of China (Grant No. 11604203).