Volume 1 Issue 10 - October 26, 2007
Softening phenomena of single-walled carbon nanotubes at higher temperature
Jin-Yuan Hsieh 1, Jian-Ming Lu 2, 3, Min-Yi Huang 2 and Chi-Chuan Hwang 2,*

1Department of Mechanical Engineering, Ming Hsin University of Science and Technology, Hsinchu 304, Taiwan
2Department of Engineering Science, National Cheng Kung University, Tainan 701, Taiwan
3National Center for High-Performance Computing, No. 28, Nanke 3rd Road, Sinshih Township, Tainan County 741, Taiwan
* Corresponding author. E-mail: chchwang@mail.ncku.edu.tw
Paper published in Nanotechnology, Vol. 17, No. 15, 14 August 2006, pp. 3920-3924 (2006)

Figure 1 The structures of carbon family.(http://cohesion.rice.edu/naturalsciences/smalley)。
Carbon nanotubes are an epoch-making important discovery after the discovery of buckminsterfullerenes by Kroto and Smally in 1985. In 1991, the carbon nanotubes were successfully synthesized by Dr. Sumio Iijima, a Japanese electron microscope investgator in the fundamental research laboratory of NEC Company. They were found in the form of needlelike, hollow and multi-layered structures at the negative plate while being synthesized by the arc discharge method. Through the high resolution transmission electron microscopy, they are identified as tubes with diameter in the range of nanometers to ten nanometers and length in micrometers. The allotrope of carbon family includes diamond, graphite, buckminsterfullerene and carbon nanotubes as shown in figure 1. Due to low density, superb flexibility, high heat conductivity and high electric conductivity, even though nanometers in size, carbon nanotubes are an ideal quasi-one dimensional quantum system. Many researchers in the world are attracted to devote themselves to perform research and development related carbon nanotubes.

Figure 2 A SWNT can be rolled by a sheet of graphite, for example the armchair type SWNT.
In figure 2, a single-walled carbon nanotube (SWNT) is a structure rolled from a sheet of graphene. It can be divided into three kinds, such as armchair, zigzag and chiral type, according to its structure. However, it can also be divided into three categories, including metal, narrow-gap semiconductor and moderate-gap semiconductor for its electronic property, according to its rolling direction. The helix of a SWNT causes the six rings of a SWNT’s wall distorted, and makes unsaturated double-bonds on the rings of a SWNT to provide resistance while electrons being conducted. Hence, differences in the electronic property of a SWNT occur. As for manufacturing carbon nanotubes, the arc discharge method is the earliest technique. However, at present time, the most efficient way is chemical vapor deposition methods. The price of carbon nanotubes, more expensive than gold at present, will drop when high quality and mass-production can be well-controlled in manufacturing processing. Low price of carbon nanotubes makes unlimited progress in the industries, and facilitates that are involved with the development of fuel cell, composite materials, semiconductor devices, field-emission flat panel displays, heat pipe, wireless communication and hydrogen green energy etc.
Figure 3 Various applications of carbon nanotubes.

In present study, the Young’s modulus of a SWNT has been obtained by employing molecular dynamics (MD) simulation numerical method. Our results, as shown in figure 4, reveal that the Young’s modulus of a SWNT increases with decreasing tube radius. Previous researchers investigated the Young’s modulus of SWNTs reveal a constant trend when the tubes’ radii are nearly equal to others. Therefore, the Young’s modulus of a SWNT is independent on its tube’s length, temperature and chiral vector. However, it holds true under relative lower temperature, but it softens beyond 1000 K.
Figure 4 The thermal noise model of a SWNT
Many experimental and numerical results inferred that the Young’s modulus of a SWNT is between 0.95~5.5 TPa, and approaches that of a sheet of graphene. We employ MD simulations, that is suitable to simulate nanoscale thermal noise of a SWNT, and utilize the classical vibration formula of macroscale cantilever as shown in equation (1) to investigate the Young’s modulus of a SWNT. We succeed in predicting reasonable Young’s modulus of a SWNT by examining its chiral vector, diameter, length and temperature. Our results are in agreement with previous studies.

Figure 5 Variation of σ2 as function of L3 for (14, 0) SWNT at temperatures of T=500K (short dash), 1000K (long dash), and 1500K (solid), respectively.

Through the standard deviation σ of the vibrating cantilever, we are able to describe the transverse amplitude u of a cantilever beam of a SWNT under constant temperature. In the equation (1), L is the length of a SWNT, k the Boltzmann constant, Y the Young’s modulus, δ the tube wall thickness (δ = 0.34 nm) and R the tube radius. In this study, the standard deviation results are calculated on the basis of more than 500 periods of the tube’s thermal vibration. The interaction between Young’s modulus and diameter, length, and temperature can be investigated in a similar manner. In equation (1), the Young’s modulus of a SWNT can be inversely obtained from the parameters of temperature T, the tube’s length L and the tube’s radius R. Here the term 1+(δ/2R) 2 can be regarded as a constant between 1 and 2, and can be calculated since σ2 and δ are given. Figure 5 indicates that the Young’s modulus of a SWNT has no relationship with the tube’s length form the variation of σ2 as function of L3.

Figure 6 shows that the Young’s modulus of a SWNT increases with decreasing the tube’s diameter, and approaches gradually that of a sheet of graphene when the tube’s diameter increases. Figure 7 indicates that the Young’s modulus of a SWNT has no relationship with the chirality structure of a SWNT. The Young’s modulus of a SWNT is a constant at lower temperatures, but that decreases sharply with increasing temperature beyond 1000K, and hence the SWNT softens. However, the trend alleviates when the temperature approaches 1800K and beyond. Even though the Young’s modulus of a SWNT decreases, it is still higher than other materials, in the range of 1~1.5 TPa. Further, from statistical analysis, figure 8 shows that the variance of the angular correlation function is used to understand why the SWNT’s soften. Moreover, we can confirm that 1100K is the transformation temperature of a SWNT.
Figure 6 The distribution curve of Young’s Moduli from (7,0) to (29,0) shown on the curve fitted by hollow circles. In the top left of the figure, they are simulated by the results including quantum effect. The dotted line indicates the Young’s modulus of graphene. Others results are from previous experimental or simulated data.
Figure 7 The Young’s Modulus of a SWNT is independent of its structure, as shown (5,5) and (9,0) nearly overlapped. The Young’s Modulus of a SWNT is a constant under lower temperature. It is independent of its length, temperature and chiral vector, however, it soften beyond 1000K.

Figure 8 The structural transformation temperature of a SWNT, 1100K, can be verified by the angular correlation function. In the figure, the solid line indicates the relation between the angular correlation function and temperature of a (7,0). The long dash line indicates that of (14,0) and the short dash line does that of a (21,0), respectively.

The new finding confirms that the softening phenomena of a SWNT at higher temperature and the transformation temperature of a SWNT. Our results utilize the macroscale experimentally measurable parameters, including temperature, the length, the amplitude of thermal noise and the SWNT’s diameter, to accurately predict the Young’s modulus of a SWNT, that is nanoscale and in the immeasurable realm.
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