本文以分子動力學研究石墨烯帶的楊氏係數及破壞應力與螺旋性和寬度尺寸的關係，研究結果發現，扶手椅型與鋸齒型石墨烯帶之楊氏係數皆有明顯的尺寸效應，但趨勢相反，且隨著寬度增加趨近於一個值；扶手椅型石墨烯帶破壞應力在寬度大於4.6nm時無明顯尺寸效應，在寬度小於4.6nm時有明顯的上升；鋸齒型則是沒有明顯尺寸效應，主要與邊界原子排列型態有關。扶手椅型石墨烯帶破裂線與自由邊界夾角約60°，鋸齒型破裂線則與自由邊界夾90°。當石墨烯帶在受外部均勻應變下，內部的原子間變形會有應變不均勻的情況，觀察破裂時的鍵長以及鍵角變化，發現破裂的產生與鍵長是否達到一臨界值有相當大的關係，當石墨烯帶內部的原子間鍵長達1.83Å左右時會開始產生破壞，但破裂的起始位置則與瞬間的原子動能分布有關。
另外本研究以分子動力學結合振動分析方法，研究石墨烯和可調式奈米碳管之共振行為，藉由快速傅立葉轉換處理分子模擬中輸出之數據進行分析。主要著重於初始狀態對石墨烯片頻率的影響以及不同組合的可調式碳管頻率。結果發現石墨烯片頻率會受初始模態位移的大小所影響，故以施加初始模態取得頻率時要注意所施位移的大小；而不同組合之可調式奈米碳管頻率會有所不同，內外管的間距在2.03~3.39Å時，將巨觀連體力學的提摩仙可樑振動理論加入修正長度可以預測碳管右端頻率。
關鍵字：奈米碳管、分子動力學、快速傅立葉轉換、振動分析

In this study, molecular dynamics simulation was employed to investigate the size and chirality effects on Young's modulus and failure stress of graphene sheet. From the results, it was found that Young's moduli of both armchair and zigzag graphene sheets showed obvious size dependence but with opposite trend. The failure stress of armchair graphene sheet possessed size effect for width less than 4.6nm, but the failure stress of zigzag one did not show obvious size effect. The failure grew along a broken line about 60° (90°) with free boundary for armchair (zigzag) graphene. It was observed that the internal atomic deformation was not uniform due to the discrete atomic configuration even though the applied external strain was uniform. It was noted that there existed a critical bond length of 1.83Å beyond which the graphene sheet would start to break under tensile strain. However, the starting failure point would depend upon the atomic kinetic energy.
Moreover, the resonance behaviors of graphene and adjustable double-walled carbon nanotubes (DWCNTs) were studied using molecular dynamics combined vibration analysis. The temporal information from molecular simulation was analyzed using fast Fourier transform. We focused on the influence of the initial modal displacement on the resonance frequency of the graphene sheet and investigated how different geometry condition would affect the frequency of the adjustable DWCNTs. It was found that the magnitude of the initial modal displacement would alter the resonance frequency of the graphene sheet. Hence, we should pay some attention to the initial modal displacement in determining the vibrational behavior of nanostructures. It was also observed that the resonance frequencies of the adjustable DWCNTs could be reasonably predicted by Timoshenko beam theory with properly selected material parameter s and corrected length.

Keywords: carbon nanotubes, molecular dynamics, fast Fourier transform, vibration analysis

[1] Y. Son, M. L. Cohen and S. G. Louie, “Energy Gaps in Graphene Nanoribbons,” Physical Review Letters, Vol. 97, pp. 216803, 2006.
[2] Z. Xu, “Graphene Nano-Ribbons under Tension,” Journal of Computational and Theoretical Nanoscience, Vol. 6, pp. 625-628, 2009.
[3] H. Bu, Y. Chen, M. Zou, H. Yi, K. Bi and Z. Ni, “Atomistic Simulations of Mechanical Properties of Graphene Nanoribbons,” Physics Letters A, Vol. 373, pp. 3359-3362, 2009.
[4] Z. Ni, H. Bu, M. Zou, H. Yi, K. Bi and Y. Chen, “Anisotropic Mechanical Properties of Graphene Sheets from Molecular Dynamics,” Physica B, Vol. 405, pp.1301-1306, 2010.
[5] J. S. Bunch, A. M. van der Zande, S. S. Verbridge, I. W. Frank, D. M. Tanenbaum, J. M. Parpia, H. G. Craighead and P. L. McEuen, “Electromechanical Resonators from Graphene Sheets,” Science, Vol. 315, pp. 490-493, 2007.
[6] A. Sakhaee-pour, M. T. Ahmadian and R. Naghdabadi, “Vibrational Analysis of Single-Layered Graphene Sheets,” Nanotechnology, Vol. 19, pp. 2085702, 2008.
[7] J. R. Mianroodi, S. A. Niaki, R. Naghdabadi and M. Asghari, “ Nonlinear Membrane Model for Large Amplitude Vibration of Single Layer Graphene Sheets,” Nanotechnology, Vol. 22, pp.305703, 2011.
[8] K. Iyakutti, V.J. Surya, K. Emelda and Y. Kawazoe, “ Simulation of Ripples in Single Layer Graphene Sheets and Study of Their Vibrational and Elastic Properties,” Computational Materials Science, Vol. 51, pp. 96-102, 2012.
[9] J. Cumings and A. Zettl, “Low-Friction Nanoscale Linear Bearing Realized from Multiwall Carbon Nanotubes,” Science, Vol. 289, pp. 602-604, 2000..
[10] Q. Zheng and Q. Jiang, “Multiwalled Carbon Nanotubes as Gigahertz Oscillators,” Physical Review Letters, Vol. 88, pp. 045503, 2002.
[11] K. Jensen, C. Girit, W. Mickelson and A. Zettl, “Tunable Nanoresonators Constructed from Telescoping Nanotubes,” Physical Review Letters, Vol.96, No. 215503, 2006.
[12] K. Jensen, J. Weldon, H. Garcia and A. Zettl, “Nanotube Radio,” Nano Letters, Vol. 7, pp.3508-3511, 2007.
[13] J. W. Kang, Y. G. Choi, Y. Kim, Q. Jiang, O. K. Kwon and H. J. Hwang, “The Frequency of Cantilevered Double-Wall Carbon Nanotube Resonators as a Function of Outer Wall Length,” Journal of Physics-Condensed Matter, Vol. 21, pp. 385301, 2009.
[14] J. W. Kang, K. R. Byun, O. K. Kwon, Y. G. Choi and H. J. Hwang, “Gigahertz Frequency Tuner Based on a Telescoping Double-Walled Carbon Nanotube: Molecular Dynamics Simulations,” Molecular Simulation, Vol. 36, pp.418-424, 2010.
[15]C. M. Wang, Y. Y. Zhang and X. Q. He, "Vibration of Nonlocal Timoshenko Beams," Nanotechnology, Vol. 18, pp. 105401, 2007.
[16] J.-C. Hsu, R.-P. Chang, and W.-J. Chang, "Resonance Frequency of Chiral Single-Walled Carbon Nanotubes using Timoshenko Beam Theory," Physics Letters A, Vol. 372, pp. 2757-2759, 2008.
[17] J. H. Irving and J. G. Kirkwood, “The Statistical Mechanical Theory of Transport Properties. IV. the Equation of Hydrodynamics,” Journal of Chemical Physics, Vol. 18, pp. 817-823, 1950.
[18] R. J. Arsenault and J. R. Beeler, Computer Simulation in Material Science, Asm International, USA, 1988.
[19] F. Reif, Fundamentals of Statistical and Thermal Physics, McGraw Hill, pp. 48-49, 1985.
[20] C. L. Tien and J. H. Lienhard, Statistical Thermo Dynamics, MeiYa, pp. 211-233, 1971.
[21] J. M. Haile, Molecular Dynamics Simulation, John Wiley and Sons, New York, 1997.
[22] C. Goze, P. Bernier and A. Rubio, “Elastic Properties of C and BxCyNz Composite Nanotubes,” Physical Review Letters, Vol. 80, pp. 4502-4505, 1998.
[23] G. C. Maitland and M. Rigby, Intermolecular Forces, their Origin and Determination, Oxford University Press, London, 1987.
[24] D. C. Rapaport, The Art of Molecular Dynamics Simulation, Cambridge University Press, 1997.
[25] 朱訓鵬，國立成功大學機械工程研究所博士論文，2002.
[26] J. E. Lennard-Jones, “The Determination of Molecular Fields II. from the Equation of State of A Gas,” Proceedings of the Royal Society, Vol. 106A, pp. 441, 1924.
[27] J. Tersoff, “New Empirical Model for the Structural Properties of Silicon,” Physical Review Letters, Vol. 56, pp. 632-635, 1986.
[28] J. Tersoff, “New Empirical Approach for the Structure and Energy of Covalent Systems,” Physical Review B, Vol. 37, pp. 6991-7000, 1988.
[29] J. Tersoff, “Empirical Interatomic Potential for Carbon with Application to Amorphous Carbon,” Physical Review B, Vol. 38, pp. 9902-9905, 1988.
[30] N. Miyazaki and Y. Shiozaki, “Calculation of Mechanical Properties of Solids Using Molecular Dynamics Method,” The Japan Society of Mechanical Engineers, Vol. 39, pp. 606-612, 1996.
[31] J. Cooley and J. Tukey, “An Algorithm for the Machine Computation of the Complex Fourier Series,” Mathematics of Computation, Vol. 19, pp.297-301, 1965.
[32] Bruel and Kjaer, “Structural Testing Part2 : Modal Analysis and Simulation,” pp.5, 1988.
[33] W. Zhao, F. Wu, H. Wu and G. Chen, “Preparation of Colloidal Dispersions of Graphene Sheets in Organic
Solvents by Using Ball Milling,” Journal of Nanomaterials, Vol. 2010, pp. 528235, pp. 1-5, 2010.
[34]C. W. Fan, J. H. Huang and C. Hwu, “Mechanical Properties of Single-walled Carbon Nanotubes - A Finite Element Approach,” Vol. 33-37, pp.937-942, 2008.