Atomistic-Continuum Study of the Mechanical Properties and Behaviors of Carbon Nanotubes

以原子-連體力學模型研究奈米碳管機械性質與行為

劉楊倫(Yang-Lun Liu)

指導教授 ： 陳文華 ； 鄭仙志

國立清華大學/工學院/動力機械工程學系/博士(2010年)

摘要

An explosive growth in research has excited an extraordinary degree of interest in material science on finding their advanced physical properties/behaviors and engineering applications ever since the discovery of carbon nanotubes (CNTs). Due to the reported exceptionally remarkable physical/mechanical properties, such as low density, high aspect ratio, high stiffness, high strength, and excellent electrical/thermal properties, they have shown great potential for a wide range of applications. Despite of the claimed advantageous technical features, it is essential to have a thorough and clear comprehension of them prior to the realization and enhancement of their engineering application.
A recent development in computational methods based on equivalent continuum modeling (ECM) has allowed an effective and efficient characterization and simulation of the mechanical properties and behaviors of a larger scale of nano-structured systems in a longer time span. In principle, the ECM approach, such as the molecular structural mechanics (MSM) model, transforms chemical bonds between atoms in molecular mechanics into a continuum model using finite element (FE) methods. However, there are certain technical insufficiencies and shortcomings in the existing ECM approach, such as the incapability of handling the surface and temperature effects. Thus, the main goal of the study is to develop an advanced ECM model that can take into account the surface and temperature effects for assessing the axial/radial mechanical properties and dynamic/static behaviors of CNTs. The study starts from the derivation of the axial mechanical properties of CNTs. The focus is placed on the study of the influences of the surface effect and the in-layer non-bonded van der Waals (vdW) atomistic interactions on the mechanical properties of CNTs. To achieve the goal, an atomistic-continuum modeling (ACM) approach is proposed. The ACM approach incorporates molecular dynamics (MD) simulation for simulating the initial relaxed unstrained configuration due to the surface effect, and an improved MSM model for simulating the effect of the in-layer vdW interactions on the mechanical properties and behaviors of CNTs. In the improved MSM model, the covalent bonds between two linking carbon atoms are simulated by a pseudo-round beam element while the associated non-bonded interactions by a non-linear spring element. Next, the temperature effect is also taken into consideration in the present modeling through Badger’s rules together with Nos-Hoover thermostat for constant temperature MD simulation. The effectiveness of the ACM approach is demonstrated through the comparison with the theoretical/experimental data available in literature. By the approach, the axial Young’s modulus, shear modulus, and vibrational/buckling behaviors of CNTs are calculated, where the influences of the in-layer vdW interactions, temperature and surface effect are also discussed thoroughly.
The study also attempts to explore the radial mechanical properties and behaviors of CNTs. In view of the technical disadvantages of the MSM model, such as the theoretical overestimate of the minor axis bending stiffness of the cross section of the covalent bonds in CNTs, thereby potentially leading to an inaccurate prediction of the corresponding radial mechanical properties, a modified MSM model is introduced. The proposed model is derived through a major modification on the bending stiffness of the covalent bond in the minor principal centroidal axis of the section based on the inversion energy in molecular mechanics. Specifically, the interactions between two carbon atoms are modeled with continuum pseudo-rectangular beams, rather than round ones, based on the second generation force field while the non-bonded vdW interactions among atoms are simulated with spring elements based on the Lennard-Jones (L-J) potential. The pseudo-rectangular beam consists of a different bending rigidity along the two principal centroidal axes of the cross section, the stiffness parameters of which are estimated based on the bond-angle variation energy and the weak inversion energy in molecular mechanics. By the approach, the radial mechanical properties of CNTs, such as the radial modulus, radial buckling load and radial deformation, are calculated. To validate the proposed MSM model, the present results are compared with those of the original MSM model and the published theoretical and experimental data.
At last, the modified MSM model is also extended to the study of the radial breathing mode (RBM) frequencies and mode shapes of CNTs. The focus of the study is placed upon exploring the characteristics and differences of the RBM/RBM-like modes of the CNTs resulted from their large diameter-length aspect ratio and/or asymmetric atomic configuration, and also, their temperature effect. By the approach, their dependence on the layer number, aspect ratio and chirality of the CNTs and temperature is also assessed. The validity of these calculated results is extensively confirmed through comparison with the literature theoretical and experimental data.
The present simulation results demonstrate several fundamental nano-effects and physical phenomenona in CNTs, which are valuable for further fundamental researches or applications of CNTs. It is worth mentioning that the applicability of the proposed methodology would not be limited to an SWCNT system or CNT system but can be extended to other nanosystems.

關鍵字

奈米碳管 ； 原子-連體模型 ； 分子力學 ； 分子結構力學 ； 凡得瓦力 ； 機械性質/行為 ； 徑向呼吸模式

並列摘要

自1991年奈米碳管被發現後，許多研究者針對其先進的物理性質與工程應用進行深入且廣泛的研究。由於奈米碳管具有優越的物理與機械性質，如低密度，高深寬比，高強度，以及特殊的電/熱性質，使其在各方面的應用具有極高的潛力。而為了實現其實際的工程應用，吾人需對其基本的物理與機械性質有清楚且充分的瞭解。
在最近幾年發展的計算方法中，等效連體力學模型（Equivalent Continuum Modeling, ECM）可針對具有大尺度之奈米結構的機械性質與行為進行有效且快速的分析與計算。基本上，ECM方法，如分子結構力學(Molecular Structure Mechanics, MSM)模型，主要是將原子間的化學鍵轉化為一等效的連體模型，再利用有限單元法(Finite Element Method)進行計算。然而，在現有的ECM方法中仍有些不足之處，如無法考慮表面/溫度效應(Surface/Temperature Effect)等。因此，本論文主要的目的為提出一更先進的ECM模型，其中在探討奈米碳管之機械性質與行為時可考慮表面/溫度之效應。本論文首先提出一原子-連體模型(Atomistic-Continuum Modeling, ACM)方法以探討表面效應與層內凡得瓦力(in-layer van der Waals (vdW) interaction)對奈米碳管之軸向機械性質的影響。而ACM方法主要包含分子動力學(Molecular Dynamics, MD)與改良型MSM模型，分別模擬由表面效應所造成的初始變形以及層內凡得瓦力。在改良型MSM模型中，原子間的鍵結力與非鍵結力分別由一等效的圓形樑單元以及一非線性彈簧單元來模擬。接著，溫度效應則由Badger’s rules以及定溫MD 模擬搭配Nos-Hoover熱容法。此ACM方法之正確性主要由文獻中之理論及實驗的結果來驗證。由此方法可計算出奈米碳管之軸向楊氏模數(axial Young’s modulus)，剪切模數(shear modulus)，以及振動/挫曲性質，其中並討論層內凡得瓦力，溫度以及表面效應之影響性。
此外，本論文也針對奈米碳管之徑向的相關機械性質進行探討。由於在原始MSM模型中，等效樑單元之斷面均假設為圓形，然而此假設使得樑單元在徑向之彎曲強度過強，造成其徑向機械性質的計算錯誤。因此，吾人提出修正型MSM模型，主要根據分子力學(molecular mechanics)中之反轉能(inversion energy)來計算等效樑單元在徑向之彎曲強度。其中，原子間之鍵結力改由矩形樑來模擬，而凡得瓦力則由彈簧單元根據Lennard-Jones (L-J)勢能函數來模擬。由此方法，可計算出奈米碳管徑向之機械性質，如徑向模數，徑向挫曲，以及徑向變形。為了驗證此修正型MSM模型，吾人將現有的結果與文獻中之實驗以及理論之結果作比較。
最後，此修正型MSM模型也進一步用來探討奈米碳管之徑向呼吸模式(Radial Breathing Mode, RBM)之頻率與模態。其主要探討RBM與類RBM(RBM-like)振動模態之特性以及其中之差異。由此方法並可得到奈米碳管之層數，長寬比，幾何形式，以及溫度對RBM頻率之影響。所得到結果也與許多文獻之結果作比較。
本論文所得到的結果顯示出許多奈米碳管中的奈米效應及物理性質，此結果相當有助於奈米碳管之基礎研究及其應用。本文所提出的模型也可更進一步運用在其他奈米材料之各種物理性質的探討。

並列關鍵字

Carbon Nanotube ； Atomistic-Continuum Modeling ； Molecular Mechanics ； Molecular Structural Mechanics (MSM) ； van der Waals Interaction ； Mechanical Properties/Behaviors ； Radial Breathing Mode (RBM)

參考文獻

[1] Abrams, Z. R., and Hanein, Y., 2007, “Radial deformation measurements of isolated pair of single-walled carbon nanotubes,” Carbon, Vol.45, pp.738-743.

[2] Agrawal, P. M., Sudalayandi, B. S., Raff, L. M., and Komanduri, R., 2008, “Molecular dynamics (MD) simulations of the dependence of C-C bond length and bond angles on the tensile strain in single-wall carbon nanotubes (SWCNT),” Computational Material Science, Vol. 41, pp.450-456.

[3] Allinger, N. L., 1977, “MM2. A Hydrocarbon Force Field Utilizing V1 and V2 Torsional Terms,” Journal of the American Chemical Society, Vol. 99, pp. 8127-8134.

[4] Allinger, N. L., Yuh, Y. H., and Lii J. H., 1989, “The MM3 force field for hydrocarbons,” Journal of the American Chemical Society, Vol. 23, pp. 8551-8582.

[5] Badger, R. M., 1934, “A relation between internuclear distances and bond force constants,” Journal of Chemistry Physics, Vol. 2, pp. 128-131.