本文利用有限元素法結合統計熱力學之關係式，可估測出單層與多層奈米碳管之熱膨脹係數及比熱性質。而研究方法之主要概念是將奈米碳管結構視為空間衍架結構，並將奈米碳管結構中的碳原子、共價鍵及凡得瓦爾力，分別利用有限元素法軟體中的質量元素、樑元素及彈簧元素來等效，並結合量子力學中用來描述原子間勢能之modified Morse potential與材料力學之樑理論，來求取等效樑元素的非線性機械性質；而凡得瓦爾力則由Lennard-Jones potential來得到其非線性彈簧勁度關係。利用這些非線性元素可以建立出更加貼近實際奈米碳管結構之有限元素分析模型，而後進行各種估測過程所需之有限元素分析，如模態分析、靜態分析等等，再結合統計熱力學中的配分函數，連結分析結果與熱力學性質之對應關係，藉此研究各種不同碳管結構的熱膨脹係數與比熱性質，並將研究結果與相關文獻相互比較，顯示本文之估測數值與現行其他理論方法所得到之數據大致吻合，並佐證本文之研究方式之可行性與準確性。

SUMMARY
This paper links the finite element analysis method and the statistical thermodynamic together to estimate the coefficient of thermal expansion (CTE) and specific heat (Cv) of carbon nanotubes. The structure of carbon nanotubes is looked as some kind of truss structure, and the material properties of the element of this truss structure could be find out from the constants of molecular dynamics. The computational results indicated that both of the axial and radial CTE of single wall carbon nanotubes (SWNTs) are positive and increasing with increasing temperature. The radial CTE of nanotubes will increase with increasing diameter of nanotubes. The radial CTE of double wall carbon nanotubes (DWNTs) is negative at low temperature range. And the CV of nanotubes is increasing with increasing temperature too. The effect of the nanotube chirality is slight both on CTEs and CV.
Key words: carbon nanotube, thermal expansion, specific heat.
INTRODUCTION
The carbon nanotubes is well known for its good strength properties when it was be found at 1990 by Sumio Iijima. The other of properties of nanotubes such as conductivity and thermal properties are also tested at the following research. The CTE and Cv of nanotube will be very important when the nanotubes may be used as the material of integrated circuit in the future. As such, there are many papers discuss about them. Bandow, S. used the X-ray diffraction technique to measure the CTE of multi-wall carbon nanotubes (MWNTs) in the temperature range 10-320K. Maniwa et al. also used X-ray diffraction to find the radial CTE of MWNTs is in the range of "1.6-2.6×" 〖"10" 〗^"-5" " " "K" ^"-1" . There are no experimental report about the SWNTs because the difficult to get individual SWNTs and design an appropriate experiment.
Besides experimental method to measure the CTE and Cv, numerical simulation is the another way that used by many papers. Raravikar et al. investigated the CTE of (5,5) and (10,10) SWNTs by molecular dynamics (MD) simulations. Schelling and Keblinski also use MD and lattice dynamics calculation to studies the CTE of other kinds of carbon structure, such as graphite, diamond and CNTs.
METHODS
The statistical thermodynamic connects the micro physical with macro thermal properties by partition function. When we assume the two carbon atoms at CNTs molecular are a pair of harmonic oscillator, we could use the vibration partition function to describe the energy between these two atoms. The unknown variables in the vibration partition function are the vibration frequency of harmonic oscillator. Therefore, as we could simulate the nature frequencies of entire equivalent CNTs truss structure, the coefficient of thermal expansion and specific heat of nanotubes could be estimated.
We use modified Morse potential and Lennard-Jones potential to describe the covalent bond between carbon atoms and the van der Waals’ force between the layers of nanotube. Both of these two potentials nonlinearly vary with the distance between two atoms. We use the commercialized finite element method software-ANSYS to help us to simulate the nature frequencies and the deformation of CNTs under pressure. The elements chose to simulate the mass of atoms, the covalent bonds and the van der Waal’s force are Mass21, Beam188 and Combin39 separately in ANSYS. During the CNTs model building process, the real radius of CNTs is derived specially to replace the function value which will cause some deviation when calculate the radius of small CNTs and make sure that the model would like to the real CNTs as close as possible.
RESULTS
At this chapter, we will discuss the result of our work and compare with the others papers. Different from the common concept that the CTE of item will be a constant value, the CTE of CNTs will vary with the temperature. We find that the effect of the chirality of CNTs is slight to both of axial and radial CTE. Both of axial and radial CTE will be converge when the length of CNTs is long enough. The axial CTE is almost not change with the increasing diameter of CNTs, but the radial CTE will. This is because the structure along the axial to support the longitudinal pressure is the same. On the other hand, the ability of the CNTs to bear the radial pressure is weaker when the diameter is increasing. That leads to significant deformation of CNTs. As we know the diameter is the most effectively factor to CTE change, we test a set of MWNTs with same outside diameter but different the number of layers and find out that the radial CTE of MWNTs will decreasing with increasing layer.
The specific heat of CNTs is also changing with the temperature. The value of specific heat is almost the same of all kind of CNTs.
CONCLUSION
The coefficient of thermal expansion and specific heat of many kinds of CNTs are estimated successfully by using the statistical thermodynamic. Although the absolute value has some different between other papers, the values of CTE are in the similar range. So we could sure that the method we used is work to estimate the thermal properties of carbon nanotubes. In the future, nanoscale carbon material may be used in many region depended on the special strength and thermal properties.

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