Temperature difference-powered carbon nanotube bearings

Abstract

Molecular dynamics simulations are conducted to study the motion of carbon nanotube-based nanobearings powered by temperature difference. When a temperature difference exists between stator nanotubes, the rotor nanotubes acquire a higher temperature, which arises from the interaction between phonon currents and nanotubes. The thermal driving force increases with the increase in temperature difference between the stators, an increase that is nearly proportional to the temperature difference. Confined by the minimum energy track, the (5, 5)@(10, 10) nanotube bearings only translate along the axis direction but without successive rotation.

Keywords

temperature difference-induced motion / carbon nanotubes / nanobearing / molecular dynamics simulation

Introduction

Technological progess has made it possible to manipulate matter at micrometer and nanometer scales. The design and creation of microelectromechanical and nanoelectromechanical systems have thus attracted much attention[1]. Of particular interest are carbon nanotubes which pose a great advantage in the fabrication of micro/nano devices because of their unique properties. Using carbon nanotubes, researchers have designed nanodevices such as nanobearings [2], nanogears [3], nanoswitches [4], and nanoscillators [5].

The bearing is an elementary part in machines. Its motion and driving force are important factors for machine design. A carbon nanotube-based bearing is usually power-driven by an electrical field [6]. Recently, the thermal gradient (temperature difference) has been demonstrated to be capable of powering solid nanodevices [7-11]. Schoen et al. simulated nanoparticle motion inside carbon nanotubes driven by thermal gradients, and found that the nanoparticles always move to the cold ends of the nanotubes [7,8]. Zambrano et al. [10] and Shiomi et al. [11] also observed analogous phenomena in their studies about the motion of water droplets inside carbon nanotubes. Barreiro et al. [9] observed the relative motion of multi-walled carbon nanotubes induced by thermal gradients in their experiments. In all the systems above, the fixed parts have successive temperature profiles, and the mobile parts, which are very short compared with the fixed parts, move to the cold ends all the time. The mechanism of the motion of nanotube based-bearing in which the mobile parts are longer than the fixed parts and the fixed parts are discrete has not been discussed before. Based on this idea, this paper studies the possibility of carbon nanotube-based bearings being powered by temperature difference.

Simulation system and method

The modeling system is shown in Fig. 1. The inner nanotube layer is a (5, 5) carbon nanotube (rotor) with a length of 25 nm. The outer layer consists of two (10, 10) carbon nanotubes (stators) with a length of 2.5 nm. The distance between the stators is 17.5 nm.

In the simulations, the intrawall C-C bonds are modeled by the bond-order potential developed by Brenner [12]:

(1)E_{\mathrm{b}}={\displaystyle \sum _i{\displaystyle \sum _{j(>i)}{f}_{ij}(r_{ij})[V_{\mathrm{R}}(r_{ij})-b_{ij}{V}_{\mathrm{A}}(r_{ij})]}},

where E_b is the total potential, r_ij is the distance between atom i and atom j, and V_R(r) and V_A(r) denote the pair-additive repulsive and attractive interactions, respectively. They have the pair-potential form as follows:

(2)V_{\mathrm{R}}(r)=A_{\mathrm{1}}{\mathrm{e}}^{-{\eta }_{\mathrm{1}}r},

(3)V_{\mathrm{A}}(r)=A_{\mathrm{2}}{\mathrm{e}}^{-{\eta }_{\mathrm{2}}r},

where A₁, A₂, η₁, and η₂ are potential coefficients. f_ij is the cutoff function which limits the distance of the bond

(4)

where b_ij represents a many-body coupling between the bond from atom i to atom j and the local environment. Interwall interactions between the carbon atoms occur via the Lennard-Jones potential in the form of

(5)V(r)=\mathrm{4}ϵ\left[{\left({\displaystyle \frac{\sigma }{r}}\right)}^{\mathrm{12}}-{\left({\displaystyle \frac{\sigma }{r}}\right)}^{\mathrm{6}}\right],

in which σ=4.204 meV and ϵ=0.337 nm.

The original bond length is 1.44 nm, and the free boundary condition is used at the two ends of the system. The Verlet leap frog scheme is applied to integrate the motion equations with a time step of 0.5 fs. The first 12.5 ps is used to maintain the system at the average temperature T₀. Next, the Nose-Hoover thermostats are attached to the two stators to maintain them at different temperatures T_l and T_r, respectively. Afterward, 60 ps is used to make the system move to a stationary state, during which atom velocities in the inner layer are rescaled to restrict the rotor from translation or rotation. The restriction is then removed and the motion result of the rotor is recorded for about 200 ps.

Results and discussion

For comparison, nine cases were simulated in which temperature differences between the two stators are positive, negative, or zero. The nine cases are tabulated in Table 1. The position of the rotor as the function of time in each case is demonstrated in Fig. 2. Although small temperature differences can also be used to study the phenomena, large temperature differences were selected in the MD simulations to save on simulation time and to reduce fluctuations.

When the temperature of the two stators are at both 400 K (no temperature difference), the rotor only oscillates along the x axis direction randomly with a small amplitude. When the temperatures of the stators are different, however, the rotor’s motion is not random, but has direction bias. This phenomenon indicates that, just like other driving forces, temperature difference can also be used to power the motion of the nanobearings. After the rotor passes through a distance (about 2.3 nm), one end of the rotor approaches the stator and the potential between the two layers leads to the reverse motion. This phenomenon is similar to what has been observed in carbon nanotube oscillators simulated by Zheng et al. [5].

Figure 2 indicates that, at the beginning, the rotor always moves to the direction of the stator at higher temperature. This directional bias is different from those observed in experiments and simulations in Refs. [7-11]. In these research, the mobile parts (outer tubes, nanoparticles, or water droplets) always move to the cold ends of the systems. This difference is caused by the difference in system structures. In other research studies, the temperature gradients exist in the fixed parts. The mobile parts, which are short compared with the fixed parts can be regarded as having uniform temperature. In our experiment, however, the temperature profiles are formed in the mobile parts while the fixed parts can be regarded as mass points and have uniform temperature.

The movement to the opposite direction, as a result of the structural difference between the fixed and mobile parts, can be explained by the mechanism of the temperature difference-induced motion. Barreiro et al. [9]and Schoen et al. [8]attributed the temperature difference-induced motion to the interaction between phonon currents (especially the breathing mode phonons) and other matter particles. The simulation system of Barreiro et al. may serve as an example[9]. When a temperature gradient exists in the inner tube, the phonon current flows from the hot part to the cold part. When the breathing mode phonons hit the outer tubes, they lose their momentum and transfer them to the outer tubes. The outer tubes are then actuated to move toward the cold parts. In our simulation systems, however, the phonon currents, which flow through the rotors from the hot ends to the cold ends, hit the cold stators, and the counteracting forces drive the rotors to move to the high temperature direction.

The rotor velocity as a function of time in each case is presented in Fig. 3. At the beginning of the motion, the velocity increases from zero and it is almost proportional to time, which is indicative of constant acceleration. This means that a constant force is applied on the rotor. The driving force is obtained by fitting the velocity and averaging the results of the cases with the same absolute temperature difference. The driving force, varying with the temperature difference, is shown in Fig. 4. As the temperature difference increases from 200 to 800 K, the driving force increases from 3.8 to 15.8 pN. In theory, besides driving force, there should be friction applied on the rotor. Servantie et al. [13,14]found that friction is linear to relative velocity in double-walled carbon nanotubes when velocity is not high. They estimated the friction coefficient of armchair-armchair double-walled carbon nanotubes to be about 6 amu/ps. At the beginning of the motion, the rotor velocity is almost less than 30 m/s, so the friction can be assessed to be smaller than 0.6 pN in our simulations. This value is much smaller than the driving force, hence it is reasonable to ignore the friction at the beginning of the motion.

When the temperature difference of stators increases, the driving force also increases, and it is approximately proportional to the temperature difference:

(6)F_T=\alpha \mathrm{\Delta }T

where the coefficient α=1.94×10^-14 N/K. This is in accordance with the observed motion of nanoparticles in carbon nanotubes simulated by Schoen et al. [7]and in the double-walled carbon nanotubes in our early study[15]. In our early research on the thermally driven motion of double-walled carbon nanotubes, the coefficient between driving force and temperature gradient depends on the nanotube structure. It varies when the chirality pairs of double-walled carbon nanotubes change. Analogously, the coefficient in the nanobearing should also be dependent on the chirality pairs of the rotors and stators. This will be studied in our future work.

Theoretically, rotors can translate or rotate. In our simulations, however, only successive translations are observed. The successive rotation may also be obtained considering other carbon nanotube chirality pairs because the motion behavior greatly depends on the interlayer potential patterns of the carbon nanotubes [15]. The rotation angle as a function of time in case 3 is shown in Fig. 5 as an example. In this process, the rotor only oscillates around an angle but does not rotate in succession. The amplitude of the rotation angle is less than 15 degrees. In other cases, the same phenomenon can be observed. This motion-confined phenomenon is related to the potential patterns between rotors and stators and has been discussed in detail in Ref. [15]. For our simulation systems, i.e., (5, 5)@(10, 10) carbon nanotubes-based bearing, the minimum energy track of potential pattern is along the axis direction. Consequently, the translation can occur easily, while the rotation is confined by the potential barrier along the circumferential direction. When the chiralities of rotors and stators change, the potential patterns also change[16,17], and the type of motion exhibited by the nanobearings will change. According to this observation, nanobearings with different types of movements or motions (e.g., translation, rotation or both of them) may be designed.

Conclusion

Using molecular dynamics simulations, temperature difference or thermal gradient is shown to be capable of being applied to drive the motion of carbon nanotube-based bearings. When a temperature difference exists in the stators, the rotor always moves to the hot end. This motion is powered by the counter force of the interaction between the phonon currents flowing through the rotors and the stators. When the temperature difference increases, the thermal driving force also increases, and it is approximately proportional to the temperature difference. At the beginning of the motion during which the friction can be ignored, the temperature-driven force is estimated to be several to tens of pN. Motion fashions of the rotors depend on the potential patterns between rotors and stators. Therefore, based on these observations, various nanobearings may be designed by choosing nanotubes with different chiralities.

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