Quantitative control of a rotary carbon nanotube motor under temperature stimulus

Due to their excellent mechanical properties [1–6] such as ultralow friction and in-shell super-strength, multi-walled carbon nanotubes (MWCNTs) have been widely adopted in constructing nanomotors and nanobearing [1, 7, 8] and strain sensors [9]. Fennimore et al [10] were the first to build a true nano-actuator by selecting MWCNTs as a key motion-enabling element. Cai et al [11] investigated self-excited oscillation triggered by nonequilibrium attraction of the ends of two tubes. In the electromechanical rotational actuator built by Bourlon et al [12], a plate was attached on MWCNTs that acted as a bearing and the actuator was driven to rotate by a charged stator. To avoid an electromigration or random walk effect, Barreiro et al [13] built a thermally driven nanomotor based on MWCNTs. In their nanomotor, an outer tube can move by rotation or translation along the axis of inner tubes when a temperature gradient exists along the axis. Motivated by the observation that electron tunneling can cause periodic vibrational [14] and translational motions in molecules [15], Wang et al [16] proposed an electron tunneling-driven rotary nanomotor in which fullerene blades were charged/discharged periodically and driven to rotate in an external electronic field. Optical [17] and chemical methods [18] have also been used to drive the rotary or translational motion of a molecular motor.

Noting that high environmental temperature induces marked thermal vibration of atoms, an easy way to actuate the motion of a molecular motor is to increase temperature. Besides the work by Barreiro et al [13], Xu et al [19] investigated the motion of a double-walled carbon nanotube (DWCNT)-based motor in a temperature field with thermal fluctuation, observing the rotation of an inner tube within a fixed tube. Guo et al [20] studied edge barriers and forces of the interlayer interaction of DWCNTs due to environmental temperature, and found that the edge force is linearly dependent on the temperature. Recently, Cai et al [21] studied the thermally driven rotation of the inner tube, including the effects of the differences between the radii of inner and outer tubes, temperature, and the length of the fixed part of the stator. The reported rotational frequency of the rotor was over 160 GHz. That work demonstrated the feasibility of a thermally driven motor made from DWCNTs. However, in the models involved in that work, neither the in-depth mechanism of uniform temperature driven rotation nor the quantitative relationship between the rotation of the rotor and the configuration of the system was addressed. In the present study, a quantitative relationship between rotor's speed and interaction energy is presented and used to investigate mechanisms of the DWCNT-based rotary motor driven by temperature stimulus. It provides a feasible way of designing nano-devices whose rotor's rotational speed is controllable.

Figure 1(a) shows a model of a nanomotor made from commensurate DWCNTs of (5, 5)/(10, 10). The two parts of the outer tube are arranged near the two ends of the inner tube. At the beginning of each simulation, the tubes are allowed to equilibrate under a Langevin-style thermostat to a specified temperature of 300 K, following energy minimization. After the minimization, the carbon atoms on the stators are fixed and the system is placed in a canonical NVT ensemble for 10 ns of simulation. The time step in the integral of Newton's equation is 1 fs. In this work, molecular dynamics simulation is carried out using LAMMPS [22], AIREBO potential [23] is used to describe the force field.

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The radial deviation degree (RDD) represents the asymmetry of the outer tube (stator in figures 1(b) and (c)), which is calculated by the formulation

$$\begin{eqnarray}&&{\rm{RDD}}{=}\displaystyle \frac{{\rm{d}}{r}}{{l}_{{\rm{c}}-{\rm{c}}}},\end{eqnarray} \tag{ 1 }$$

where ${\rm{d}}r$ is defined in figures 1(b) and (c). ${l}_{{\rm{c}}-{\rm{c}}}\ =\ 0.142\; nm$ in this work. When RDD = 0, all the end carbon atoms on the stator are in an ideal circle with the radius of r. A higher value of RDD means that there is a smaller distance between the rotor and stator.

The interaction energy is the difference between the potential of the present system and the potentials of the isolated inner tube and two outer tubes. When the atoms in stators are fixed, the interaction energy can also be considered as the increment of the rotor's potential.

From figure 2, the increment of the rotational speed of the rotor is proportional to the increment of the interaction energy of the system, i.e.

$$\begin{eqnarray}&&{\rm{\Delta }}\omega =-\alpha \cdot {\rm{\Delta }}{P}_{{\rm{r}}-{\rm{s}}},\end{eqnarray} \tag{ 2 }$$

where $\alpha$ is a coefficient depending on temperature and radial deviation of the carbon atom at the outer end of stator.

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This phenomenon reveals that the angular acceleration of the rotor is proportional to the interaction power of the system, i.e.

$$\begin{eqnarray}&&\varepsilon =\underset{{\rm{\Delta }}t\to 0}{\mathrm{lim}}\displaystyle \frac{{\rm{\Delta }}\omega }{{\rm{\Delta }}t}=-\alpha \cdot \underset{{\rm{\Delta }}t\to 0}{\mathrm{lim}}\displaystyle \frac{{\rm{\Delta }}{P}_{{\rm{r}}-{\rm{s}}}}{{\rm{\Delta }}t}.\end{eqnarray} \tag{ 3 }$$

Equation (3) implies that we can control the acceleration of the rotor's rotation by controlling the variation of interaction energy, i.e., the power, of the system in a design.

From figure 3, we find that the rotational direction of the rotor in the stators with the 'A-type' scheme is anticlockwise, due to the positive value of rotational frequency. Interestingly, if we adjust the RDD of the green atom on the stator (B-type), the rotational direction is clockwise. It should be mentioned that the hydrogen atoms are bonded with the carbon atoms on the internal ends of stators. After hydrogenation, the edge barrier of potential of the stator is reduced to be close to that of the mid part of the stator. Hence, the rotation of the inner tube is purely due to the interaction of the outer edges of tubes. As a result, we can conclude that the rotational direction (clockwise or anticlockwise) of the rotor is determined by the type of radial deviation, here, including 'A-type' or 'B-type' in figures 1 or 3. This finding implies that we can engender a specified rotational direction of the rotor by selecting the radial deviation type. We also find that the final values of the rotational speed for the two schemes are identical to ~308 GHz when they share the same RDD value.

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In figure 4(a), the histories of the rotational speed of the rotor in the system with the 'A-type' scheme (figure 1(b)) are presented at different environmental temperatures. A higher environmental temperature leads to a higher rotational speed of the rotor. At the same time, the initial value of interaction energy of the system increases with an increase in the environmental temperature (figure 4(b)), and the absolute value of the increment of interaction energy is also more obvious at a higher temperature. Hence, we conclude that a higher temperature leads to a greater decrease in the potential of the system, resulting in a higher speed of rotation of the rotor.

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When the value of RDD varies (figure 5(b)), we find that the absolute value of the increment potential of the system does not always increase with an increase in RDD. Correspondingly, the rotational speed of the rotor has the same characteristics (figure 5(a)). For example, when the RDD is no higher than 0.1, the increment of interaction energy of the system is far less than that when RDD > 0.2. In particular, the absolute value of increment approaches its peak when RDD = 0.4. The rotor's rotational speed is also the highest among the seven schemes. When RDD > 0.4, the absolute value drops. The reason for this phenomenon is that the attraction between rotor and stator increases when the atom on the stator with radial deviation is too close to the rotor. In fact, too high value of RDD leads to sudden stoppage of the rotor [24].

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From the discussion above, we know that the rotational speed of the rotor is determined by the increment of the potential energy of the system. We also know that the interaction energy depends on two factors. One is the charge number of the end carbon atoms, which is fixed in this study. The other is the distance between the rotor and the radially deviated atom on the stator, namely, both the value of gap (figure 1(a), which varies in simulation) and the RDD. Therefore, we can assume that the atoms on the rotor initially vibrate to near its equilibrium state, but the state changes when one of the atoms impacts on the radially deviated atom on the stator. During that impact, a circumferential force appears that causes the rotor to rotate. Higher potential change near the radially deviated atom leads to higher circumferential force. And some conclusions can be drawn.

• (1)  
  A higher temperature, or more drastic thermal vibration of atoms on the rotor, leads to a greater change of the potential of system, resulting in a higher speed of rotation of the rotor.
• (2)  
  Meanwhile, the type of the radially deviated atom determines the direction of the circumferential force, which is always opposite to the rotational direction (clockwise or anticlockwise) of the rotor.
• (3)  
  The absolute value of the potential increment of the system does not always increases with an increase in the RDD. For example, the absolute value approaches its peak when the RDD equals 0.4.

By adjusting either environmental temperature or RDD, the rotor with a specified rotational speed can be obtained. The rotational direction can also be adjusted by varying the RDD type.

The financial support from the National Natural-Science-Foundation of China (Grant No. 11372100) is acknowledged.