利用原子連體力學法及等效方法探討多重尺度之結構分析

Analysis of Multi-scale Structures Using the Atomistic-Continuum Mechanics and the Equivalent Methods

黃昭荏(Chao-Jen Huang)

指導教授 ： 江國寧

國立清華大學/工學院/動力機械工程學系/博士(2012年)

摘要

隨著奈米科技不斷地發展與提升，許多研究學者為了探討奈米結構材料性質紛紛提出各式模擬方法，而常見分析法包含量子力學、分子動力學、蒙地卡羅法等。然而這些方法受限於電腦計算能力，故無法有效探討大尺度奈米結構。因此，如何降低中央處理器的計算時間，且同時保有原材料性質，即為重要議題。
為了降低大尺度奈米結構的計算時間及複雜度，本研究以原子連體力學法建立等效結構模型。其中，原子連體力學法是由原子力學、連體力學、等效理論、有限單元法與高速計算理論所組成，利用此方法預估多重尺度結構的機械強度。原子連體力學法是將原本不連續的原子結構等效成連續體結構，過程中不需假設鍵結的楊氏係數與截面積即具分析能力。該法建立之結構模型可分別使用拉伸試驗或模態分析，將其分析結果之反力或振動頻率反推結構的楊氏係數。本研究延伸應用此法預估矽鍺異質結構，並探討奈米碳管複合材料特性。
在評估矽鍺異質結構材料過程中，由於矽和鍺晶格尺寸不匹配，使得矽鍺於磊晶過程中將因材料比例或厚度差異呈現出不同的變形情形。本研究利用原子連體力學法、拘束方程式搭配微¬¬-巨觀理論描述奈米尺度下矽/矽鍺/矽異質結構承受應變的情形，進而探討矽與鍺在不同比例與不同厚度的情形下對於整體結構應變改變程度，將其結果與文獻比對驗證該法可行性。本研究分析結果符合文獻所提及當結構長度在50 nm時，將因邊緣的彎矩使整個應變矽承受壓應變的狀況，因此該分析法可供奈米尺度下應變矽在半導體元件尺度的設計方針。
奈米碳管楊氏係數藉由原子連體力學法分析得知，不論拉伸或模態分析所得之楊氏係數與文獻提供實驗結果具一致性，故此原子連體力學法的模型可有效描述奈米碳管結構之機械性質。本研究基於原子連體力學所得奈米碳管結構特性，分別使用等效固體元素、薄殼元素、臂元素的模型取代。當等效模型外加負載為拉伸、扭轉或剪力下反應出之物理行為與原子連體力學物理行為相近時，即稱其為有效的等效模型。這些等效結構模型可使用比原子連體力學法更少的元素數目描述相同的物理行為，並藉此克服電腦運算能力極限，以提升結構計算尺度。此外，由於迄今仍尚未有文獻明確定義奈米碳管束楊氏係大小是否與碳管束直徑相關。因此本研究提出三種較常被採用的奈米碳管束截面積方式探討其楊氏係數，結果顯示奈米碳管束實驗過程中選擇以外接圓方式定義其截面積為主流。此截面積假設指出楊氏係數隨著碳管束直徑增大後逐漸趨於穩定數值，此趨勢是由於每根碳管所圍成的面積及碳管間的餘隙面積比值的影響，當碳管束直徑增大時該比值隨著逐漸達穩定值所反應出的結果。
針對局域至廣域分析應用，本研究利用酚醛樹脂與奈米碳管束複合材料為例。驗證代表性體積元素其拉伸方向楊氏係數理論解與數值解的一致性，進而討論加強材料隨機分佈時影響楊氏係數的情形。本研究結果得知，等效模型可同時維持材料特性並減少計算上所需耗費的時間。透過與文獻驗證後，確認等效方法可有效使用於不同材料，並可著手從微觀至巨觀或巨觀至微觀的觀點進行探討分析。

關鍵字

原子連體力學法 ； 有限單元法 ； 等效方法 ； 多重尺度結構 ； 楊氏係數 ； 奈米碳管 ； 複合材料

並列摘要

The continuous development and improvement in the nanotechnology field prompt many researchers to develop various simulation methods to determine the material properties of nanoscale structures. The most common simulation methodologies include Quantum Mechanics, Molecular Dynamics, and Monte Carlo, among others. However, these methods are restricted by the time limitation of the central processing unit (CPU) computer hardware, which cannot estimate larger-scale nanoscale models. Thus, decreasing the CPU processing time and retaining the physical properties of nanoscale structures have become critical issues.
To decrease the CPU processing time and complexity of larger nanoscale models, the current study utilized atomistic-continuum mechanics (ACM) to build an equivalent model. ACM consists of atomic mechanics, continuous mechanics, equivalent theory, finite element method, and high-speed computing theory to estimate the mechanical properties of a multi-scale structure. ACM transfers an originally discrete atomic structure into an equilibrium continuum model, and does not require the assumption of the Young’s modulus and the cross-sectional area of each chemical bond. ACM can allow the Young’s modulus to be obtained using the same model for tensile and modal analyses. This study investigates the mechanical properties of silicon (Si)-germanium (Ge), hereafter SiGe, heterostructures and carbon nanotube (CNT) composite materials.
In the estimation of the material properties of SiGe heterostructures, different heterostructure volume fractions and thicknesses reflect the different deformations caused by the Si lattice constant that is not equal to that of Ge. This study utilized the ACM method, constraint equation, and local-global theory to establish a conceptual framework that links the lattices of Si and Ge. Therefore, this strategy can describe the strain effect caused by the lattice mismatch in the nanoscale heterostructure. The strain distribution with Si and Ge having different volume fractions and different thicknesses is investigated. The analytical result is also validated with previous studies indicating that the entire top Si layer surface depicts compressor strain when the mesa length is 50 nm. This study establishes a simulation method to obtain the mechanical behavior of nanoscale strained-silicon and serve as a guide for semiconductor devices design.
The Young’s modulus of CNTs can be presented using the ACM method. Both tensile and modal analytical results agree with the experimental results in literature indicating that the ACM model can properly describe mechanical properties. Based on this result, this study investigated the equivalent solid, shell, and beam models to generate similar mechanical behaviors with the ACM model. The similar mechanical behavior of the equivalent model includes the model under tensile, torsion, or shear external loading. These equivalent models can significantly reduce the required total element number and CPU processing time to investigate a larger nanoscale structure. This study also adopted three cross-sectional area definitions to explore whether the Young’s modulus of CNT ropes depends on the cross-sectional area definition. The results indicate that the Young’s modulus distribution based on the circumcircle assumptions well agrees with most of the experimental results. Hence, most experimental methods adopted the circumcircle to obtain the Young’s modulus of the CNT ropes. The circumcircle assumption involves the distribution of the tubes and the gap between each tube. The ratio between the gap and tube areas becomes a stable value when the diameter of the CNT ropes is increased. Therefore, a larger diameter of CNT ropes that represents the Young’s modulus becomes a stable value, as mentioned in literature.
This study adopted phenolic resin/CNT composite material to discuss local and global technique applications. The representative volume element was utilized to validate the consistency of the Young’s modulus of theoretical and numerical results. The equivalent models simultaneously decrease the CPU processing time and maintain mechanical behavior, making them sufficient, accurate, and acceptable. This equivalent method is feasible from a local to global perspective and vice versa.

並列關鍵字

atomistic-continuum mechanics method ； finite-element method ； equivalent method ； multi-scale structural ； Young’s modulus ； carbon nanotubes ； composite material

參考文獻

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