Effect of defects on the motion of carbon nanotube thermal actuator

3.1 Single atom vacancy defect

In order to investigate the effect of single atom vacancy defect on the movement behavior of nanotube, the following

three defect types are used, 2 vacancy defects adjacent, 2 vacancy defects opposite, 4 vacancy defects. The variation of position, velocity of the outer nanotube versus time during the process are shown in Figures 5(a) and (b) respectively. It can be seen from Figure 5(a) and (b) that the presence of single atom vacancy defects reduces the movement magnitude and increases the cycle time, which indicates that energy dissipation and damping effect are introduced to the system by the single atom vacancy defects. It shows that the extent of damping is different for different situations. The system with 4 defects undergoes the largest reduction in magnitude and also the largest increase in cycle time, while less damping effects are observed in the systems with 2 defects opposite and 2 defects adjacent. The degree of the resistance is largely dependent on the number and distribution of the defects.

Figure 5

Variation of the movement behaviors with time for the system containing no defect, 2 vacancy defects adjacent, 2 vacancy defects opposite,4 vacancy defects in the inner nanotube: (a) The position with time. (b) The velocity with time

The external force in the direction of movement of the outer tube fluctuations during the first move period are shown in Figure 6(a)-(d), It is seen clearly that, in the case of no defect, the external force fluctuation range is small, leading to small energy dissipation.However, the presence of single atom vacancy defect causes a large fluctuation of the external force, which causes larger energy dissipation. From Figure 6(b)-(d), it can be seen that due to the region of defects is localized at the center of the inner nanotube, the most significant energy loss occurs only when the outer nanotube passes through that region, which can be seen from the two points A and B in the figure. During the first move period, there are three forces in the system. F_T is the driving force induced by the thermal gradient, f is the friction and f_d is the friction caused by the defects. To study the effect of the defects on the motion of the outer tube, we ignore the friction f here. The driving force has been demonstrated to be proportional to the system temperature [19], and it is a constant because of the stable temperature. So the fluctuation of force in point A and B are caused by the defects.At point A, the outer tube overcomes the energy barrier generated by the defect, and the defect performs negative work on the outer tube, which acts as resistance. And at point B, the outer tube is about to leave the defect, the defect performs positive work on the outer tube, and the external force is expressed as thrust. From Figure 6(a)-(d), we can roughly calculate the amount of resistance caused by defects in various situations. The average value of external force in the absence of defects is 0.08630 eV/nm. While in other cases, when the outer tube is far from the defect, the average value of the external force is equal to the average value of the external force in system without the defect approximately. The magnitude of the resistance calculated from the external force diagrams is as follow, The damping in the system with 2 vacancy defects adjacent, 2 vacancy defects opposite and 4 vacancy defects is 0.4482eV/nm, 0.4068 eV/nm and 0.8449 eV/nm respectively. This calculation result is consistent with the result obtained from the position and velocity diagrams. It can therefore be concluded that the damping force follows the order of f_{4 vacancy defects} > f_{2 vacancy defects adjacent} > f_{2 vacancy defects opposite} > f_{no defect}.

Figure 6

Comparison of the external force acting on the outer tube during the first period of movement for the double-walled system containing no defect, 2 vacancy defects adjacent, 2 vacancy defects opposite,4 vacancy defects in the inner nanotube. The direction of the external force points to the positive direction of the coordinate

To understand the discrepancies in energy dissipation of the different cases, the rotational speed with respect to time of the outer nanotube is calculated. From Figure 7, we can see that the defect-free system achieves a stable rotational speed in the short start of the movement. While in the system containing defect, we can see that the introduction of defects changes the direction of the rotational speed, and the rotational speed of the outer tube increases each time the defect passes. We can find that the trajectory of the outer tube is not just a straight line along the central axis of nanotube, but also undergoes the self-rotation motion. The introduction of vacancy defect damages circular symmetry of the inner tube. The potential energy barriers originate in the asymmetry configuration. The first principles calculations [28] showed that the formation energy of a single atom vacancy defect is 7.6 eV. The presence of persistent rotation motion leads to persistent energy dissipation, eventually the outer nanotube is unable to pass through the energy barrier.

Figure 7

Rotational frequencies of outer tube for the double-walled system containing no defect, 2 vacancy defects adjacent, 2 vacancy defects opposite, 4 vacancy defects in the inner nanotube

3.2 Stone–Wales defect

Next, we study the cases where the different number of Stone–Wales defects are distributed around the inner nanotube. Unlike the single atom vacancy defect, in the situation of system with Stone–Wales defect, the centroid circumference of inner nanotube is deformed and the deformations can be observed obviously. The following three defect types are used, 2 SW defects adjacent, 2 SW defects opposite, 4 SW defects. The graphs of position, velocity of the outer nanotube versus time during the process are shown in Figures 8(a) and (b), respectively. It can be seen from Figure 8(a) and (b) that the presence of SW defects reduces the movement magnitude and increases the cycle time, which indicates that energy dissipation and damping effect are introduced to the system by the SW defects. It shows that the extent of damping is different for different situations. The system with 4 defects undergoes the largest reduction in magnitude and also the largest increase in cycle time, while less damping effects are observed for the systems with 2 defects opposite and 2 defects adjacent. The degree of the resistance is largely dependent on the number and distribution of the defects.

Figure 8

Variation of the movement behaviors with time for the system containing no defect, 2 SW defects adjacent, 2 SW defects opposite, 4 SW defects in the inner nanotube: (a) The position with time. (b) The velocity with time

The external force in the direction of movement of the outer tube fluctuations during the first move period are shown in Figure 9(a)-(d), It is seen clearly that, in the case of no defect, the external force fluctuation range is small, leading to small energy dissipation. However, the presence of SW defect causes a large fluctuation of the external force, which causes larger energy dissipation. From Figure 9(b)-(d), it can be seen that due to the region of defects is localized at the center of the inner nanotube, the most significant energy loss occurs only when the outer nanotube passes through that region, which can be seen from the two points A and B in the figure. As described earlier, during the first move period, there are three forces in the system, and the fluctuation of force at point A and B are caused by the defects. From Figure 9(a)-(d),we can roughly calculate the amount of resistance caused by defects in various situations. The average value of external force in the absence of defects is 0.08630 eV/nm. While in other cases, when the outer tube is far from the defect, the average value of the external force is approximately equal to the average value of the external force without the defect. The magnitude of the resistance calculated from the external force diagrams is as follow: The damping in the system with 2 SW defects adjacent, 2 SW defects opposite and 4 SW defects is 0.4812 eV/nm, 0.5132 eV/nm and 0.9567 eV/nm respectively. This calculation result is consistent with the result obtained from the position and velocity diagrams. It can therefore be concluded that the damping force follows the order of f_{4 SW defects} > f_{2 SW defects opposite} > f_{2 SW defects adjacent} > f_{no defect}.

Figure 9

Comparison of the external force acting on the outer tube during the first period of movement for the double-walled system containing no defect, 2 SW defects adjacent, 2 SW defects opposite,4 SW defects in the inner nanotube. The direction of the external force points to the positive direction of the coordinate

To understand the discrepancies in energy dissipation of the different cases, we also calculate the rotational speed with respect to time of the outer nanotube. From Figure 10. However, we found that the effect of SW defects on the speed of rotation is not obvious, the SW defects do not change the direction of the rotation speed, and the influence on the magnitude of the amplitude is not obvious. The effect of the SW defect on the system stems from its high energy barrier. The presence of the circumference-distributed SW defects gives rise to a potential barrier when the outer nanotube telescopes through the inner nanotube. Depending on the number of SW defects distributed along the nanotube circumference, the energy barrier profile and magnitude of the defect cluster will be different. The formation of an SW defect involves the rotation of a pair of carbon atoms from the circumferential direction to the axial direction. As a result, the number of carbon atoms in the circumferential direction is reduced, and the nanotube contracts radially at the defected region. This deformation also affects the inner nanotube motion behavior in two ways. On one hand, the first principles calculations [28] showed that the formation energy of an SW defect is 9.2 eV. It exerts an energy barrier at the defect location that the inner nanotube must overcome in order to pass through. On the other hand, the self-rotation motion caused by deformation also leads to energy dissipation as the inner nanotube passes through the site of SW defect, eventually the outer nanotube is unable to pass through the energy barrier.

Figure 10

(a) Rotational frequencies of outer tube for the double-walled system containing no defect, 2 SW defects adjacent, 2 SW defects opposite, 4 SW defects in the inner nanotube

3.3 The effect of temperature and chirality

In the discussion above, we only considered the case where the average system temperature is 300K, but did not consider the situation at other average temperatures. But system average temperature is very important to this thermal actuator. So we also consider the case where the system temperature is 250K and 350K and molecular dynamics were used to simulate the motion of the systems with no defect and the systems with different kind of defects. The variation of position of the outer nanotube versus time during the process in the system temperature of 250K are shown in Figures 11(a) and (b) respectively. And The variation of position of the outer nanotube versus time during the process in the system temperature of 350K are shown in Figures 12(a) and (b) respectively. Comparing Figure 11 and 12 with Figure 5 and 8, it can be found that when the system temperature increases from 250K to 350K, the energy loss of the system increases. Further, the reason for

Figure 11

Variation of the movement behaviors with time for the system in 250K and 300K: (a) The system containing no defect, 2 vacancy defects adjacent, 2 vacancy defects opposite, 4 SW defects in the inner nanotube, (b) The system containing no defect, 2 SW defects adjacent, 2 SW defects opposite, 4 SW defects in the inner nanotube

Figure 12

Variation of the movement behaviors with time for the system in 350K and 300K: (a) The system containing no defect, 2 vacancy defects adjacent, 2 vacancy defects opposite, 4 SW defects in the inner nanotube, (b) The system containing no defect, 2 SW defects adjacent, 2 SW defects opposite, 4 SW defects in the inner nanotube

this phenomenon is that the higher the temperature, the greater the interlayer friction between the inner CNT and the outer CNT, so the energy loss is larger.

It is well known that chirality has a great influence on the properties of carbon nanotubes, so we have further considered zigzag chirality and perform molecular dynamics simulation. The added model is that the thermal actuator system consisting of a (9, 0) inner nanotube with length of 60 nm and a (18, 0) outer nanotube with length of 6 nm. The results are very surprising. Under the seven conditions we simulated: no defect, 2 vacancy defects adjacent, 2 vacancy defects opposite, 4 vacancy defects, 2 SW defects adjacent, 2 SW defects opposite and 4 SW defects, there was only one case (2 vacancy defects opposite) where the outer tube could move in translation, while in other cases, the outer tube could not move. And The variation of position and the velocity of the outer nanotube versus time during the process in the case of 2 vacancy defects opposite of two kinds chirality are shown in Figures 13(a) and (b) respectively. From the figure we can see that although the outer tube of the zigzag chirality carbon nanotube with 2 vacancy defects can move in translation, the time to start moving is much longer than the system of armchair chirality, and the speed of movement is much slower. The reason for this phenomenon is that in armchair–armchair DWNTs, the nanotube axis direction is

Figure 13

Variation of the movement behaviors with time for the system of armchair chirality and zigzag chirality containing 2 vacancy defects: (a) The position with time. (b) The velocity with time

an easy direction of movement, while for zigzag–armchair or zigzag–zigzag DWNTs, the circumferential direction is an easy direction [27]. Therefore, in the thermal actuator system, the occurrence of the translational movement of the outer tube is related to the chirality of the inner and outer tubes.

3.4 Oscillation

In the process of studying various cases of the SW defect, we found an interesting phenomenon. In the system with 2 SW defects opposite, the outer tube can oscillate in the vicinity of the defect. Finally, we turn our attention to the oscillation in the system containing the 2 SW defects opposite. The factors that caused the oscillation phenomenon are studied. From the above simulation results, when the conditions other than the defect type are the same, only the system with the 2 SW defects opposite is oscillated, so we can see that the occurrence of the oscillation phenomenon is related to the type of the defect.

Then we change the initial position of the outer tube and placed the center of mass of the outer tube at 10 nm and 16 nm from the left boundary of the inner tube for molecular dynamics simulation. The results obtained are shown in Figure 14. It can be seen from the position diagram that when the initial position of the outer tube is p⁰ = −20nm, the outer tube is accelerated by the temperature gradient force for a longer distance, the kinetic energy of the system is higher, and the outer tube can pass

Figure 14

Variation of the movement behaviors with time for the initial position of the outer tube in system is p⁰ = −20nm, p⁰ = −14nm and p⁰ = −17nm respectively: (a) The position with time. (b) The velocity with time

the defect to the front of the defect three times, but cannot cross the energy barrier the fourth time, so the oscillation phenomenon cannot be generated, and when the initial position of the outer tube is p⁰ = −14nm, the outer tube is accelerated by the temperature gradient force for a shorter distance, and the kinetic energy of the system is lower, so that the outer tube can only cross the defect to the front of the defect once, when the second time the outer tube passes through the defect, it cannot cross the energy barrier and therefore cannot produce oscillation phenomenon. While when the initial position of the outer tube is p⁰ = −17nm, the outer tube crosses the defect to the back of the defect twice. When the third time the outer tube passes through the defect, it cannot cross the energy barrier, and the temperature gradient force acts to form an oscillation phenomenon. The above results indicate that the generation of the oscillation phenomenon is related to the initial position of the outer tube.

We also change the temperature gradient of the system and set the temperature gradient to 4.2 K/nm, 2.5 K/nm, 5 K/nm. The simulation results are shown in Figure 15. It can be seen from the figure that when the temperature gradient is 5K/nm, the temperature gradient driving force is larger, so the work is more in the same distance, and the kinetic energy of the system is higher, and the outer tube can pass the defect to the front of the defect three times, but cannot cross the energy barrier the fourth time, so the oscillation phenomenon cannot be generated; And the temperature gradient is 2.5K/nm, the temperature gradient driving force is smaller, so the work is less in the same distance, and the kinetic energy of the system is lower, so that the outer tube can only cross the defect to the front of the defect once, when the second time the outer tube passes through the defect, it cannot cross the energy barrier and therefore cannot produce oscillation phenomenon, while when the temperature gradient is 3.4K/nm and 4.2K/nm, the outer tube crosses the defect to the back of the defect twice. When the third time the outer tube passes through the defect, it cannot cross the energy barrier, and the temperature gradient force acts to form an oscillation phenomenon. The above analysis shows that the generation of oscillations is also related to the system temperature gradient.

Figure 15

Variation of the movement behaviors with time for the temperature gradient of the system is 3.3K/nm, 4.3K/nm,5K/nm and 2.5K/nm respectively: (a) The position with time. (b) The velocity with time

In summary, the occurrence of the oscillation phenomenon is related to the initial position of the outer tube, the type of the defect, and the magnitude of the temperature gradient.