Wavelike deformation traveling on a carbon nanotube

Abstract

The propagation of wavelike ripples on a carbon nanotube (CNT) induced by the radial impact of a C₆₀ molecule is investigated by molecular dynamics simulations. The ripples start at the impact point and spreads through the tube, accompanied by energy transfer. The ripples would effectively reduce the local energy concentration around the impact point. The propagation of ripples is clearly affected by the diameter but is independent of the chirality of the CNT. Noticeable diffraction occurs when the deformation ripples encounter obstacles or narrow slits, which indicates that the propagation of ripples can be used to detect defects in the CNT. This work provides new and exciting possibilities for CNTs to serve as energy buffers, sensors and new nanoelectromechanical devices.

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1. Introduction

Carbon nanotubes (CNTs) have gained intensive research interest due to their remarkable electrical, mechanical, and thermal properties.^1–4 Their high-tensile strength along the axis enables them to possess plenty of potential applications such as in super-strong fibers and novel nanometre-scale devices.^5–9 However, CNTs are very soft and susceptible to external perturbations in their radial directions. Evidence of the softness of CNTs in the radial direction has been reported in that a critical hydrostatic pressure of only 4 GPa could lead to an irreversible collapse of single-walled CNTs (SWCNTs) with a diameter of 14.08 Å.¹⁰ Riedo et al. have measured the radial stiffness and Young modulus of multiwalled CNTs (MWCNTs).¹¹ They found that the Young modulus steeply declined with an increase in the radius down to an asymptotic value of 30 ± 10 GPa for CNTs with radii larger than 5 nm. On the theoretical side, classical molecular dynamics (MD) simulations have been used to examine the radial collapse of SWCNTs and demonstrate that the collapse pressure is dependent on the diameter but independent of chirality.¹² Interestingly, the radial collapse occurring at one end of a SWCNT with an appropriate diameter (>3.5 nm) would cause a successive collapse of the tube from this end to the other one until the tube is fully collapsed, which is the so-called ‘domino effect of CNTs’.¹³

Due to the softness in the radial direction, CNTs exhibit unique deformation characteristics when subjected to bending and torsion, featuring some nearly periodic wavelike ripples.^14–22 Since this kind of deformation was discovered by transmission electron microscopy (TEM),^14–16 a lot of research work has been devoted to investigating such a rippling mode of deformation and its effect on the properties of CNTs.^17–22 Arroyo and Belytschko applied a generalized local quasicontinuum simulation (QCS) to study the nonlinear mechanical response and rippling of thick MWCNTs and found that the ripples caused by bending and torsion would reduce the effective bending stiffness and torsional modulus respectively.¹⁷ Yang et al. used MD simulations to reveal a twisting mode coupled with the rippling mode while MWCNTs underwent the bending load and found that the geometrical characteristics were in good accordance with those from experiments and QCS.¹⁸ Moreover, Duan and Wang employed a twisted CNT to serve as an energy pump for the possible smooth transport of water molecules as the twisting ripples propagated along the tube.¹⁹ As bending and twisting can produce a rippling deformation of CNTs, it arouses our interest to investigate whether other mechanical action would bring about such interesting performance. In this paper, we report a study on the wavelike ripples on a CNT induced by the striking of C₆₀ molecules. Since the strain energy also transfers in the form of ripple, it is revealed that the propagation of ripples corresponds to the continuous distortion of C–C bonds. This study may provide an opportunity for comprehensive and satisfactory understanding of the mechanical behavior of CNTs subjected to radial impact.

2. Modeling and methods

The collision process is carried out by means of the classical MD method. Brenner's second generation reactive empirical bond order potential,²³ which has been widely used to study the mechanical properties of CNTs,^24,25 is employed to describe the short range bonding energies among carbon atoms. The interaction between the C₆₀ and CNT is modeled by the Lennard-Jones 12-6 potential.²⁶ The behavior of atoms is simulated by solving the equations of motion using Gear's predictor-corrector algorithm, which offers a good compromise between precision and computational time.²⁷ We set up a collision system consisting of a C₆₀ and a SWCNT (40, 40) with a length of 243.5 Å. Both ends of the CNT are fixed. Initially, the C₆₀ is placed 20 Å above the CNT and starts to move towards the center of the nanotube perpendicular to axis of the tube at a speed of 15 Å ps⁻¹. The origin of the coordinates has been set at the center of the CNT. The time step is 1 fs and simulations are allowed to run for 20000 time steps. In order to reveal the striking process clearly, the atomic coordinates in the CNT are converted into plane coordinates by unzipping the tube from the back (as shown in Fig. 1). For atom i, the plane coordinate (X,Y,Z) can be changed from the tube coordinate (x,y,z) based on the following equations:where and R is the initial radius of the CNT while x₀(i), y₀(i) and z₀(i) are the initial coordinates of atom i. According to eqn (1), the carbon atoms in the tube are all located in a plane. The displacement of an atom is divided into three parts: the radial displacement, circumferential displacement and axial displacement. ΔX and ΔY represent the circumferential and axial displacements, which hardly change the shape of tube. The radial displacement, which is perpendicular to the tube wall, is set as the Z coordinate. In order to describe the deformation of the CNT, we use positive (negative) values of the Z coordinate to distinguish whether the point is convex (concave) in the surface. The precision of all coordinates is 10⁻³ Å.

Fig. 1 Schematic representation of the C₆₀–CNT system.

3. Results and discussion

Fig. 2 clearly shows the ripple-like deformation generated by the impact of the C₆₀ molecule. In this simulation, we have never found any topological defect, such as the Stone–Wales 5-7-7-5 rings that may offer an alternative source of wrinkles.²⁸ Furthermore, no bonds have been found to break or switch during the impacting process. The C₆₀ molecule does not penetrate the tube but merely distorts its cross section to produce some ripples on the surface. As shown in the snapshots of the tube, the impact of the C₆₀ molecule locally flattens one side of the tube and then the entire tube wall near the impact point begins to ‘ring’ or oscillate.²⁹ The radial striking of C₆₀ on the CNT makes the carbon atoms around the impact site move inward, resulting in a dent. The maximum depth of the dent is 4.089 Å (a cross-sectional compression by 7.54% at 2400 fs). In the end, an obvious dent disappears and the CNT recovers its initial cylindrical shape owing to elastic forces. As the disturbance created by C₆₀ travels out from the dent, the atoms do not travel with the ripple, but only move back-and-forth and up-and-down. Continuous vibration of the atoms results in the continuous distortion of C–C bonds and makes the ripples propagate farther. Actually, the ripples are the movement of the distortion of C–C bonds. From the unfolded view of CNT, it can be clearly seen that the ripples vary with the propagation direction. For example, the ripple along the circumferential direction is more distinct than that along the axial direction. It is also interesting to note that the ripples between the axial and circumferential direction fluctuate within a negative range. The high flexibility of the CNT in the radial direction plays a very important role in the formation of ripples. If all atoms in the CNT were rigidly bound, there would not be any ripples generated. Previous experimental and theoretical results show that the band structure of CNTs can be dramatically altered by mechanical strain.^30,31 Therefore, it is suggested that the wave-like deformation may also cause some changes in the electrical properties of CNTs.

Fig. 2 The radial deformation of the CNT after impact by a C₆₀ molecule. (a), (b) and (c) are the side, front and unfolded views of the CNT, respectively. The contours represent the radial displacement in units of angstrom. (d) The distribution of strain energy in the CNT. The contours are in units of eV.

Next, we will mainly investigate the ripple propagation in the circumferential and axial directions by tracing the change in displacement of an atomic ring and atomic chain. Fig. 3(a) and (b) show the radial displacement (Z) of the atomic ring and the atomic chain respectively. The radial displacements of atoms are perpendicular to the direction of the ripple propagation. It is found that the amplitude of the ripple increases with time but decreases with the distance from the impact point. At 1500 fs, the deformation of the atomic ring and atomic chain is almost the same and is scarcely affected by the curvature. However, the situation is different as the ripple propagates. Comparing Fig. 3(a) with Fig. 3(b), we can see that the expansion rate of the dent in the atomic ring is much slower than that of the one in the atomic chain, which results in an elliptical dent in accordance with the deformation of a CNT subjected to the compression of a probe tip.³² Moreover, the patterns of ripples in these two directions are quite different. The ripples propagating in the circumferential direction, possessing obvious crests and troughs, fluctuate between positive and negative, whereas the ripples in the axial direction except the middle dent fluctuate only within a positive range. In addition, the amplitude of the ripples in the axial direction is much smaller than that in the circumferential direction. For example, except for the middle dent, the largest displacement of the atomic chain is only 0.319 Å while the largest displacement of the atomic ring reaches 1.164 Å at 2500 fs.

Fig. 3 The radial displacement of (a) the atomic ring (colored yellow in Fig. 1) and (b) the atomic chain (colored green in Fig. 1). (c) The circumferential displacement of the atomic ring and (d) the axial displacement of the atomic chain.

The axial displacement of the atomic ring and the circumferential displacement of the atomic chain are so tiny that they can be neglected. Therefore, we mainly focus on the circumferential displacement of the atomic ring and the axial displacement of the atomic chain as shown in Fig. 3(c) and (d). These two kinds of displacement are both parallel to the direction of ripple propagation. At the beginning, atoms in the middle of both the atomic ring and the atomic chain tend to move away from the impact point. However, as the disturbance spreads out, there are a lot of differences between the axial and circumferential directions. For the atomic chain, except for the atoms in the middle, the other atoms are pulled towards the impact point. As shown in Fig. 3(d), there are two extrema on each side of the impact point, which represent the largest inward and outward displacement of the atomic chain respectively. Over time, the extremum representing the largest outward displacement gets smaller and smaller as the C₆₀ molecule moves away from the tube wall. Finally, atoms in the atomic chain all move towards the impact point. But for the atomic ring, the case is different. More and more atoms around the impact point are pushed to move outwards over time. In addition, the displacement of the atomic ring increases with time and is much larger than that of the atomic chain. To sum up all of the types of displacement, it is found that the circumferential and the axial displacement of atoms are both smaller than their radial displacement, indicating that it is the bending of C–C bonds rather than the stretching mode dominating the propagation of the disturbance generated by the radial impact.

Next, we use (30, 30), (40, 40) and (50, 50) CNTs to investigate the effect of diameter on the ripple propagation and use (50, 30), (40, 40) and (70, 0) CNTs to investigate the influence of chirality. As shown in Fig. 4, the radial displacement of the atomic ring and atomic chain at 2500 fs is used to analyze the ripple propagation on CNTs with different diameters and chiralities. It is found that the propagation of ripples is clearly affected by the diameter (Fig. 4(a) and (b)) but is independent of the chirality of the CNT (Fig. 4(c) and (d)). The effect of diameter on the ripple propagation in the axial direction is quite different from that in the circumferential direction. In the circumferential direction, the ripples propagating on the CNT with a smaller diameter will lag behind those on the larger diameter tube while it is not the case in the axial direction.

Fig. 4 Dependence of ripple propagation on the diameter and chirality of the CNT. (30, 30), (40, 40) and (50, 50) CNTs have the same chirality but different diameters. (50, 30), (40, 40) and (70, 0) CNTs have the same diameter (54.80, 54.24 and 54.80 Å, respectively) but different chiralities. (a) and (c) represent the radial displacement of the atomic ring at 2500 fs; (b) and (d) represent the radial displacement of the atomic chain at 2500 fs.

To reveal the energy variation of the impact process, Fig. 5 shows the strain energy and kinetic energy of the CNT as a function of time. When the C₆₀ molecule strikes the target position on the CNT, the atoms around the impact point are pushed away from their initial position by the C₆₀ molecule owing to the van der Waals interaction. The movement of atoms would inevitably lead to a change in the C–C bond length and bond angle, which results in a further change in the strain energy of the CNT. The kinetic energy of C₆₀ is transformed into the strain energy and kinetic energy of the CNT gradually. With the deceleration of the C₆₀ molecule, the energy transfer between the C₆₀ and CNT fades away. The transformation between the strain and kinetic energy of the CNT maintains a balance, which dominates the energy distribution in the tube. The refined strain energy distribution is unfolded into a flat sheet for better visibility as shown in Fig. 2(d). The energy transfer brings about some wave-like isoenergetic strips which have also been observed in the torsion of CNTs.³³ At the beginning, only a few of the atoms near the impact point possess high strain energy. The main route of the strain energy delivery in the form of a ripple is along the circumferential rather than the axial direction. It is found that the atoms buckling outward always possess higher strain energy than the atoms bucking inward. As the strain energy spreads through the CNT, the ‘strong’ impact energy is transformed into the ‘soft’ ripple energy, which effectively avoids local energy concentration. This suggests that the CNT could be used as an energy buffer to withstand impact loads.

Fig. 5 The time dependence of the energy and the dent depth.

To measure the transport of kinetic energy, we define the temperature T as 1/2mv_i² = 3/2k_BT which is also referred to as the kinetic temperature. We study the propagation of the kinetic energy with the distribution of the kinetic temperature in the above used atomic ring and atomic chain. As shown in Fig. 6, the impact of the C₆₀ molecule forms a heat pulse with a peak of 959.6 K at the impact point at 1240 fs. As the heat pulse spreads out, the peak undergoes a fast decay as an exponential function. It is found that the decline of the peak along the circumferential direction is more rapid than that along the axial direction. At 1600 fs, the heat pulse peak of the atomic ring has decreased to 62.7 K while the peak of the atomic chain only decreases to 155.6 K. Fig. 6 also illustrates the progression of the kinetic temperature. The propagation speed could be determined from the spatial distance traversed by the kinetic temperature during a given time interval. Because the locations of the kinetic temperature front are linear with time, the propagation speed of the kinetic temperature could be derived from the slope of the fitting line which is about 19.5 km s⁻¹. In SWCNT, there are three acoustic phonon modes contributing substantially to the heat propagation: transverse acoustic modes (TA), longitudinal acoustic modes (LA) and twisting modes (TW). The heat propagates at a speed of 19.5 km s⁻¹ corresponding well with the speed of a LA phonon mode.³⁴

Fig. 6 Temporal and spatial distribution of the kinetic temperature in the atomic chain. The inset shows the time dependence of the peak kinetic temperature along the axial and circumferential directions.

Because of diffraction, a wave can change direction when it encounters an obstacle or a narrow slit, which is an important property of waves. Since the ripples propagate like waves, will the ripple be diffracted when encountering obstacles or narrow slits? Here, we use the C₆₀ molecule to strike a CNT with two fixed areas where the atoms cannot move. The fixed areas serve as obstacles on the side wall to obstruct the propagation of ripples. When the deformation ripples encounter the fixed areas, diffraction occurs as shown in Fig. 7(a) and (b). The diffraction of ripples would lead to the vibration of the atoms behind the obstacles, though these atoms will lag behind other atoms in the same wavefront, which causes an apparent bending of the diffracted ripple. Although there is some diffraction or bending of the ripple around the obstacle, there is still a ‘zone of silence’ left behind it. The amplitude of the diffracted ripple is also smaller than the ripple in the same wavefront. Fig. 7(c) illustrates the distribution of strain energy in the CNT. It is found that the strain energy ripples are also diffracted around the obstacles accompanied by the diffraction of deformation ripples.

Fig. 7 Snapshots of ripple propagation on the CNT with two fixed areas (3 Å × 5 Å) which are colored black in (a). (a) and (b) are the side and unfolded views of the CNT, respectively. The contours represent the radial displacement in units of angstrom. (c) The distribution of strain energy in the CNT. The contours are in units of eV.

The above results arouse our interest in detecting the diffraction of ripples encountering narrow slits. Here, two long barriers with slits are placed on the CNT as shown in Fig. 8. Most of the ripples are held back by the fixed atomic chain. Only a few of the ripples spread out past the narrow slit resembling some semicircles surrounding the slit. Based on the Huygens–Fresnel principle, every point on a wavefront is viewed as a new point source for a secondary radial ripple. When the ripple reaches the narrow slits, the atoms in the slits vibrate like point sources, and the subsequent propagation of all these radial ripples will form a new wavefront, leading to continuously propagating ripples. From the distribution of strain energy, it can be seen that the strain energy ripples reflect at the fixed atomic chain and interfere with the incident ripples.

Fig. 8 Snapshots of the ripple propagation on a CNT with two fixed atomic chains which are colored black in (a) and (b). There is one slit (3 Å × 5 Å) on each atomic chain (all carbon atoms on the atomic chain, except those belonging to the slits, are fixed during the simulation). (a) and (b) are the side and unfolded views of the CNT, respectively. The contours represent the radial displacement in units of angstrom. (c) The distribution of strain energy in the CNT. The contours are in units of eV.

Besides the diffraction of ripples mentioned above, it is also important to study how the ripples propagate on a CNT with a vacancy. Due to the absence of atoms, there is no medium for the ripples to propagate onward and the vacancy obstructs the traveling of the wavelike deformation firstly as shown in Fig. 9. Then, diffraction occurs and makes the ripples bypass the vacancy. However, the shape of ripples would be different. Similar to the ripples encountering obstacles, the diffracted ripple appears to bend. Fig. 10 shows snapshots of the ripple propagation on the CNT with a rectangular vacancy ring. When the ripple propagates to the rectangular vacancy ring, the ripples can propagate onwards only through the graphene ribbons, which generates four new ripples along different directions. The atoms surrounding the vacancy always possess larger displacements than other outside atoms due to the lack of constraint at the side of the vacancy. Just like the ripple propagation on the perfect CNT, the diffraction ripples also vary with direction. The propagation of ripples along the axial direction is different from that along the circumferential direction. Moreover, it can be seen that the strain energy is restricted to the region around the impact point by the vacancy ring. Only a little strain energy transfers outwards through the graphene ribbons in the form of ripples.

Fig. 9 Snapshots of ripple propagation on a CNT with a vacancy. (a) and (b) are the side and unfolded views of the CNT, respectively. The contours represent the radial displacement in units of angstrom. (c) The distribution of strain energy in the CNT. The contours are in units of eV.

Fig. 10 Snapshots of ripple propagation on a CNT with a rectangular vacancy ring which separates the atoms around the impact point from the others. There are four narrow graphene ribbons with width 3 Å connecting the two parts. (a) and (b) are the side and unfolded views of the CNT, respectively. The contours represent the radial displacement in units of angstrom. (c) The distribution of strain energy in the CNT. The contours are in units of eV.

4. Conclusions

The propagation of wavelike ripples on a CNT induced by the radial impact of a C₆₀ molecule is investigated by MD simulations. It is demonstrated that the ripple starts at the impact point and spreads through the CNT, accompanied by a transfer of strain energy. The propagation of ripples is clearly affected by the diameter but is independent of the chirality of the CNT. The ripples would effectively reduce the local energy concentration around the impact point, which opens up an exciting possibility for CNTs to serve as an energy buffer to withstand impact. This wave-like deformation is an important practical issue in the physical and mechanical sciences, which would provide an opportunity for comprehensive and satisfactory understanding of the mechanical properties and the stability of CNTs to make full use of the tubes as high-strength and lightweight materials. Noticeable diffraction occurs when the deformation ripples encounter an obstacle or a narrow slit, which is of great significance to detect defects in the CNT. By tracing the propagation of ripples, one can use CNTs as sensors to monitor a disturbance resulting from the impact of external molecules.

Acknowledgements

We would like to acknowledge the support from the National Natural Science Foundation of China (Grant Nos. 50971081 and 50831003). We also thank the support from the Natural Science Fund for Distinguished Young Scholars of Shandong (JQ200817). This work is also supported by the Natural Science Fund of Shandong Province (ZR2009FM043), by the Ph.D. Dot Programs Foundation of ministry of education of China (No.20090131110025) and by the National Science Fund for Distinguished Young Scholars (No.2009JQ014) from Shandong University.

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