- 學位論文
修正型Nos-Hoover熱容法及其於奈米結構熱力及熱機械性質之探討
A Modified Nose-Hoover Thermostat and Its Application to the Study of Thermodynamic and Thermal-mechanical Properties of Nanostructures
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摘要
近年來低尺度奈米結構如奈米粒子、奈米線、奈米柱、奈米管等，因具有獨特的尺寸量子效應、奈米尺寸、單晶結構及低結構缺陷等優點，而享有與眾不同物理材料特性，因此已漸被應用於微奈米電子及機電系統中。過去針對低尺度奈米結構之理論及實驗研究大抵偏向機械、光學、電性等行為的探討。然因高功能及高效能的需求造成系統電子元件功率的大幅增加，更由於高攜帶性及高電性效能的要求使得元件尺寸日益微縮，因此系統電子元件走向高功率密度的趨勢。高功率密度不可避免帶來高溫，高溫可能改變電子材料之物理性質，且降低電子元件的電性品質，因此奈米材料之熱力及熱機械性質/行為對其工業應用有重大影響，例如提高微米及奈米元件之熱傳導性將有效提升其品質及結構的可靠度，因此有必要深入加以探究。 文獻上，針對奈米材料溫度特性之模擬常採用分子動力學(molecular dynamics, MD)或是量子力學 (quantum mechanics) 法搭配定原子數、定體積及定溫(NVT)熱容法，如速度縮放法(velocity-rescaling thermostat)、Berendsen 熱容法、標準型Nos-Hoover (NH)熱容法、鏈狀NH熱容法(NH chain)及“大規模” 鏈狀NH熱容法(“massive” NHC)等等。這些熱容法源自於不考慮原子間交互作用力之單原子氣體模型。不同於稀薄的氣體，強鍵結系統(如分子、晶體及固體)原子間的交互作用力並不可忽略。因此，這些熱容法並不適用於強鍵結系統的定溫分子動力學模擬。忽略原子位能即聲子的振動效應對物理系統之溫度的影響，將導致物理系統的溫度被低估，進而造成外加系統反饋過度的能量進入物理系統中，此將導致高溫時系統能量高度振盪，而造成系統穩定度變差、求解精度下降及原子鍵結提早斷裂等問題。 本論文旨在發展一套適用於強鍵結系統之新型正則叢集(canaical ensemble)或是NVT熱容法以利於定溫分子動力學計算。此熱容法衍生自傳統標準型NH熱容法，因此又稱為修正型NH熱容法。此法係基於Debye理論中所推導出的晶格振動能及零點能加入聲子(phonon)對系統的貢獻。數學證明顯示此修正型NH熱容法所求得之分隔函數(partition function)與正則叢集計算的相等，意謂以修正型NH熱容法所求得之物理量將等同於正則叢集之計算結果。此熱容法可作為研究固態/分子奈米結構(如金晶體及碳分子)從低溫(低於Debye溫度)至高溫(近相變化點)的熱力及熱機械性質的基礎。本文將以此修正型NH熱容法搭配分子動力學法或非平衡分子動力學法(nonequilibrium MD, NEMD)測試各式低尺度奈米結構(如奈米金粒子/奈米金線，奈米碳管及奈米碳球)之溫度相依熱力及熱機械性質(如振動行為、動態楊氏模數、熱傳導係數、線及體膨脹係數、熔點、比熱及常壓下高溫的相變化行為)。同時亦評估尺寸(長度及直徑)及旋度效應對低尺度奈米結構材料性質的影響，以及探討單層奈米碳管中尺寸效應對聲子(熱)傳輸現象之影響。為驗證修正型NH熱容法之有效性，研究結果除與現有的實驗及理論結果相互比較外，更將與搭配量子修正模式(quantum correction model)之傳統正則叢集分子動力學模擬以及修正型分子結構力學 (modified molecular structural mechanics)搭配Badger理論進行鍵長及鍵角溫度修正等模式之計算結果進行比對。 本論文之成果不僅可深入了解低尺度奈米結構之低溫熱力及熱機械性質及高溫相變化行為，亦可了解溫度、尺寸及旋度之效應。不但深具學術性更有助於低尺度固/分子態奈米材料之設計、開發及應用。
並列摘要
Because of the distinct size-dependent quantum effects together with nanosize, single crystal structure and minor defects, low-dimensional nanostructures, such as nanoparticles, nanowires, nanorods or nanotubes, yield remarkable physical material properties, and thus are potential for use in nano-scale electronic or electromechanical devices. Many previous theoretical and experimental studies of low-dimensional nanostructures were mainly placed on their various thermal-mechanical, optical and electronic properties. However, with the continually decreasing size of electronic and micromechanical devices and also the dense integration of both passive and active components for diversified functions and high performance, there is a general trend toward high power density, and thus high device temperature. High device temperature would potentially vary the physical properties of electronic materials and depreciate the electrical performance of electronic devices. It is now realized that the thermodynamic and thermal-mechanical properties of low dimensional nanomaterials and their temperature dependence are also important for applications. For example, enhancing the thermal conductance in micro- and nano-devices is essential to the improvement of their thermal performance, and even to their structural reliability. For constant temperature molecular dynamics (MD) or quantum dynamics (QM) simulation, several conventional NVT thermostats have been widely applied in literature, including velocity-rescaling thermostat, Berendsen thermostat, Nos-Hoover (NH) thermostat, Nos-Hoover chain (NHC) and “massive” NHC (MNHC). These thermostats are simply derived based on a monatomic gas model, where the intermolecular interactions are neglected. Unlike dilute gases, the interatomic interactions in a tightly bound system, such as molecules, crystals and solids, are not negligible; as a result, these thermostats are not adequate for use in constant temperature MD simulation of a tightly bound system. The neglect of the potential energy of atoms and so as the effect of phonons during the correlation of the physical system energy to temperature may underestimate the temperature of the physical system. Essentially, the underestimate would lead to feedback of an excessive energy into the physical system through feedback control of the external system. This would, unfortunately, further result in large fluctuation in system energy at high temperature, which potentially causes poor system stability, reduced solution accuracy and also early rupture of atomic bonds etc., an inaccurate estimate of the thermodynamic or thermal-mechanical properties of the tightly bound system, and even premature atomic bond breaking problems at high temperature. The study aims at developing a novel canonical ensemble (constant NVT) thermostat method for constant temperature MD simulation of a tightly bound system. The thermostat method is derived based on the standard NH thermostat, and is thus termed the modified NH thermostat. By the method, the thermodynamic and thermal-mechanical properties of solid/molecule nanostructures, such as Au crystals and carbon molecules, in a temperature range from low temperature (below Debye temperature) to high temperature (near phase change point) are extensively investigated. The proposed modified NH thermostat algorithm accounts for the phonon effects by virtue of the lattice vibrational and zero-point energy, derived based on the Debye theory, to accurately capture the quantum effects, particularly at temperature below Debye temperature. Proof of the equivalence of the method and the canonical ensemble is made. The modified NH thermostat incorporated with MD or nonequilibrium MD (NEMD) simulation is tested on several different low-dimensional solid/molecule nanostructures, such as Au nano-particles/nanowires, carbon nanotubes and carbon fullerenes, to characterize their temperature-dependent thermodynamic and thermal-mechanical properties, including vibrational behaviors, dynamic Young’s modulus, thermal conductivity, linear and volumetric coefficient of thermal expansion (CTE), melting point, constant volume specific heat, and also high-temperature phase transformation behaviors at atmospheric pressure. Furthermore, their size (length and diameter), lattice orientation, and chirality dependence are also assessed, and besides, the size dependence of the phonon transport phenomena in the SWCNTs from ballistic to super-diffusive is also examined. In addition to the published experimental and theoretical data, the predicted results are compared with those obtained from the MD simulation using several conventional NVT thermostats with or without quantum corrections and the modified molecular structural mechanics (MMSM) model incorporating Badger’s rule to determine the temperature-dependent bond length and angle. The achievements of this study provide a more profound understanding of not only the low temperature thermodynamic and thermal-mechanical properties of the low-dimensional solid/molecule nanomaterials and high temperature phase transformation behaviors but also their temperature, size, lattice orientation, and chirality dependences. The derived results show valuable academic contributions, and can be of much help for the design, development and applications of the low-dimensional nanomaterials.
參考文獻
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