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研究生： 邱韋倫
Qiu, Wei-Lun
論文名稱： 以分子動力學研究臨界分解剪應力在BCC與HCP結構上的適用性
The examination of the applicability of CRSS theory in BCC and HCP structure by molecular dynamics simulation
指導教授： 張怡玲
Chang, I-Ling
學位類別： 碩士
Master
系所名稱： 工學院 - 機械工程學系
Department of Mechanical Engineering
論文出版年： 2012
畢業學年度： 101
語文別： 中文
論文頁數： 132
中文關鍵詞： 分子動力學 、奈米線 、差排 、雙晶 、臨界分解剪應力
外文關鍵詞： molecular dynamics, nanowires, dislocation, twinning, critical resolved shear stress
相關次數： 點閱：87  下載：0 
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    • 本文將以分子動力學方法研究兩種晶格結構（體心立方及六方最
    密堆積）之奈米線的彈塑性行為，並以系統性的方式來觀察降伏時的
    臨界應力值受晶格排列方向及尺寸的影響，以檢驗臨界分解剪應力理
    論在奈米尺度下的適用性。
    由模擬結果發現，體心立方結構奈米線的彈性性質、臨界應力與
    臨界應變值都沒有明顯的尺寸效應，奈米線彈塑性性質與軸向晶格排
    列方向有關，觀察降伏時奈米線的外觀與內部原子的滑動，發現不同
    晶格方向的奈米線降伏機制不同，在[100]、[110]奈米線為雙晶現象，
    而在[111]奈米線發現其塑性時產生的變形機制與[100]、[110]奈米線
    截然不同，初步斷定為差排滑移所造成，且產生的滑移系統非常的雜
    亂難以區分。經計算臨界分解剪應力發現其值差距非常大，因而推論
    在奈米尺度下臨界分解剪應力理論不適用於預測雙晶現象的發生。
    而六方最密堆積結構奈米線，除了[0001]奈米線的臨界應力有明顯
    地隨著線寬尺寸加大而遞增的現象外，其餘在[01-10]、[2-1-10]奈米線
    皆無明顯趨勢，觀察降伏時奈米線的外觀與內部原子的滑動，發現不
    同晶格方向的奈米線降伏機制皆為差排滑移。而經由臨界分解剪應力
    的計算，可發現除了[0001]奈米線因加載方向造成沿次要的滑移系統
    滑動，其餘在[01-10]、[2-1-10]奈米線與作為驗證的[10-10]、[10-11]奈米
    線所計算出的臨界分解剪應力值皆非常接近，誤差值皆在5%以內，
    故推論臨界分解剪應力理論可用來預測六方最密堆積結構奈米線的
    差排產生。
    關鍵字: 分子動力學、奈米線、差排、雙晶、臨界分解剪應力

    Molecular dynamics simulations were employed to investigate the elastic and plastic behaviors of nanowires (body-centered-cubic and hexagonal-closed-packed crystal structures). The size and axial crystal orientation effects on critical stress/strain at the yielding point of the
    nanowires were systematically studied to examine the applicability of critical resolved shear stress (CRSS) theory at nanoscale.
    From simulation, both elastic properties and critical stresses/strains of the body-centered-cubic (bcc) nanowires do not show noticeable size effect. The elastic and plastic properties of bcc nanowires depend on the axial crystal orientations. From the inspection of the atomic configuration at yielding, it was observed that the yielding mechanism was different for various oriented nanowires. It was twinning for [100], [110] nanowires
    and dislocation slip for [111] ones, whose slip systems were complicated and difficult to distinguish. The calculated critical resolved shear stresses for [100] and [110] nanowires deviated significantly, which indicated that
    the CRSS theory could not apply to the prediction of twining behavior at nanoscale.
    For hexagonal-closed-packed (hcp) nanowires, only the critical stress of [0001] nanowires showed obvious size effect. It is observed that dislocation slips were the yielding mechanisms for all crystal orientated nanowires. The calculated critical resolved shear stresses for [01-10],
    [2-1-10] , [10-10],[10-11] nanowires are quite close, except for the [0001] one. It is because the slip system appeared in [0001] nanowire was secondary due to the loading direction. It is concluded that the CRSS theory could apply to the prediction of dislocation slip appearing in hcp
    nanowire as long as the slip system is primary.
    Keywords: molecular dynamics, nanowires, dislocation, twinning,critical resolved shear stress

    目錄 摘要 ..................................................III ABSTRACT ........................................... V 致謝..................................................VI 目錄................................................. VII 圖目錄................................................ XI 表目錄............................................... XVI 第一章 緒論............................................ 1 1.1 前言...........................................................................................1 1.2 文獻回顧...................................................................................2 1.2.1 材料的機械性質之文獻回顧..................................2 1.3 本文架構...................................................................................5 第二章 差排理論、雙晶變形與分子動力學理論.............. 6 2.1 差排理論與雙晶變形...............................................................6 2.1.1 體心立方結構..........................................................6 2.1.2 六方最密堆積結構..................................................6 2.1.3 晶格缺陷..................................................................7 2.1.4 差排..........................................................................7 2.1.5 體心立方結構的差排............................................9 2.1.6 六方最密堆積結構的差排....................................9 2.1.7 雙晶的變形............................................................10 2.1.8 臨界分解剪應力....................................................11 2.2 分子動力學理論.....................................................................12 2.2.1 基本理論與假設....................................................12 2.2.2 系綜觀念................................................................13 2.2.3 分子作用力與勢能函數........................................14 2.2.4 原子級應力............................................................21 2.2.5 週期性邊界與最小映像法則................................21 2.2.6 初始條件設定........................................................24 2.2.7 運動方程式............................................................25 2.2.8 截斷半徑法與Verlet表列法...................................27 2.2.9 原子位移的分析參數............................................29 第三章 體心立方結構奈米線............................. 44 3.1 分子模擬方法與流程.............................................................44 3.1.1 原子模型................................................................44 3.1.2 勢能函數的選用.....................................................45 3.1.3 塊材模擬流程.........................................................45 3.1.4 彈性係數驗證.........................................................46 3.1.5 奈米線模擬流程....................................................47 3.2 結果分析與討論.....................................................................49 3.2.1 奈米線....................................................................49 第四章 六方最密堆積結構奈米線......................... 83 4.1 分子模擬方法與流程.............................................................83 4.1.1 原子模型................................................................83 4.1.2 勢能函數的選用.....................................................84 4.1.3 塊材模擬流程........................................................84 4.1.4 彈性係數驗證........................................................85 4.1.5 奈米線模擬流程....................................................88 4.2 分析與討論.............................................................................89 4.2.1 奈米線....................................................................89 第五章 結論與未來展望................................ 114 5.1 結論.......................................................................................114 5.1.1 體心立方結構奈米線........................................114 5.12 六方最密堆積結構奈米線.................................115 5.2 未來展望...............................................................................116 参考文獻........................117 圖目錄 圖2.1 體心立方結構，(a)原子排列、最密堆積面及(b)最密堆積方向 ....................................................................................................32 圖2.2 六方最密堆積結構之原子排列、最密堆積面及最密堆積方向 .......................................................................................................32 圖2.3 簡單立方晶格的刃差排，(a)示意圖及(b)垂直於刃差排平面 上的原子排列[9]..........................................................................33 圖2.4 簡單立方晶格的螺旋差排，(a)示意圖及(b)圍繞螺旋差排的 原子排列[9]..................................................................................34 圖2.5 布格迴路，(a)環繞正刃差排及(b)環繞右螺旋差排[9] ........35 圖2.6 古典雙晶化簡圖，(a) 晶體受剪力作用前（b）晶體受剪力作 用後...............................................................................................35 圖2.7 (a) 雙晶剪變與（b）滑移剪變的差異...................................36 圖2.8 單軸分解剪應力的幾何模型..................................................36 圖2.9 二體勢能之原子間交互作用示意圖......................................37 圖2.10 凡得瓦爾力之勢能關係........................................................37 圖2.11 多體勢能之原子交互作用示意圖........................................38 圖2.12 週期性邊界條件，(a)晶胞設定情形、(b)當原子跑出模型外且xold<0及(c)當原子跑出模型外且xold<L ....................................................39 圖2.13 最小映像法則，(a)當xij<L/2、(b) xij<-L/2、(c) xij>L/2 及(d)與週期性邊界之關係..................................................................41 圖2.14 截斷半徑示意圖....................................................................42 圖2.15 Verlet鄰近表列法示意圖........................................................42 圖2.16 Verlet表列示意圖....................................................................43 圖3.1 三種不同晶格排列之奈米線示意圖，(a)[100]奈米線、(b)[110] 奈米線及(c)[111]奈米線..............................................................67 圖3.2 (a)模擬塊材[ 0 0 0 0 0]應變、 (b) 模擬塊材 [0 0 0  0 0] 應變...........................................................68 圖3.3 半徑20 Å，長度為44.8 Å之[110]奈米線受加載與卸載之應 力應變圖......................................................................................68 圖3.4 半徑20 Å，長度為44.8 Å之[110]奈米線模型進行五次模擬 之應力應變圖..............................................................................69 圖3.5 奈米線週期性長度與臨界應力關係圖，(a)[100]奈米線、 (b)[110]奈米線及(c)[111]奈米線................................................70 圖3.6 奈米線週期性長度與臨界應變關係圖，(a)[100]奈米線、 (b)[110]奈米線及(c)[111]奈米線................................................72 圖3.7 三種不同晶格排列之奈米線與塊材的楊氏係數與尺寸關係 圖...........................................................................................................72 圖3.8 奈米線半徑與臨界應力關係圖，(a)[100]奈米線、(b)[110]奈 米線及(c)[111]奈米線..................................................................74 圖3.9 奈米線半徑與臨界應變關係圖，(a)[100]奈米線、(b)[110]奈 米線及(c)[111]奈米線..................................................................75 圖3.10 三種不同晶格排列之奈米線的臨界應力與尺寸關係圖....76 圖3.11 三種不同晶格排列之奈米線的臨界應變與尺寸關係圖....76 圖3.12 半徑20 Å，長度為84 Å 之[100]奈米線在不同拉伸狀態下 之原子圖，(a)應力下降前、(b)應力下降後.............................77 圖3.13 半徑20 Å，長度為44 Å 之[110]奈米線在不同拉伸狀態下 及卸載後之原子圖，(a)應力下降前、(b)應力下降後(c)卸載至 應力為零......................................................................................77 圖3.14 半徑20 Å、長度為84 Å 之[100]奈米線在(a)應力下降前及 應力下降後不同時間下的原子圖(b)、(c)、(d) ........................78 圖3.15 半徑20 Å、長度為44 Å 之[110]奈米線在(a)應力下降前及 應力下降後不同時間下的原子圖(b)、(c)、(d) ........................79 圖3.16 滑移向量法判斷[100]奈米線的原子移動方向...................80 圖3.17 滑移向量法判斷[110]奈米線的原子移動方向....................80 圖3.18 半徑20 Å，長度為84 Å 之[111]奈米線在不同拉伸狀態下 之原子圖，(a)應力下降前及(b)應力下降後.............................81 圖3.19 滑移向量法判斷[111]奈米線的原子移動方向....................81 圖3.20 [100]、[110]奈米線-滑移向量判斷臨界分解剪應力與尺寸關 係...................................................................................................82 圖4.1 a與b軸邊長皆為35.412 Å，c軸邊長為56.148 Å之塊材模型示 意圖.............................................................................................101 圖4.2 三種不同晶格排列之奈米線示意圖，(a)[0001]奈米線、 (b)[01-10]奈米線及(c)[2-1-10]奈米線..........................................101 圖4.3 半徑20 Å，長度為44 Å之[0001]奈米線受加載與卸載之應力 應變圖.........................................................................................102 圖4.4 奈米線週期性長度與臨界應力關係圖，(a)[0001]奈米線、(b) [01-10]奈米線及(c) [2-1-10]奈米線............................................103 圖4.5 奈米線週期性長度與臨界應變關係圖，(a)[0001]奈米線、(b) [01-10]奈米線及(c)[2-1-10]奈米線..............................................105 圖4.6 三種不同晶格排列之奈米線與塊材的楊氏係數與尺寸關係 圖.................................................................................................105 圖4.7 奈米線半徑與臨界應力關係圖，(a)[0001]奈米線、(b) [01-10] 奈米線及(c)[2-1-10]奈米線.........................................................107 圖4.8 奈米線半徑與臨界應變關係圖，(a)[0001]奈米線、(b) [01-10] 奈米線及(c)[2-1-10]奈米線.........................................................108 圖4.9 三種不同晶格排列之奈米線臨界應力與尺寸關係圖........109 圖4.10 三種不同晶格排列之奈米線臨界應變與尺寸關係圖......109 圖4.11 半徑20 Å，長度為44 Å 之[0001]奈米線在不同拉伸狀態下 及卸載後之原子圖，(a)應力下降前、(b)應力下降後及(c)卸載 至應力為零................................................................................110 圖4.12 滑移向量法判斷[0001]奈米線的滑移方向.......................110 圖4.13 半徑15 Å，長度為45 Å 之[01-10]奈米線在不同拉伸狀態下 之原子圖，(a)應力下降前 及(b)應力下降後.........................111 圖4.14 半徑15 Å，長度為45 Å 之[2-1-10]奈米線在不同拉伸狀態下 之原子圖，(a)應力下降前 及(b)應力下降後.........................111 圖4.15 滑移向量法判斷[01-10]奈米線的滑移方向.......................112 圖4.16 滑移向量法判斷[2-1-10]奈米線的滑移方向........................112 圖4.17 三種不同晶格排列之奈米線-滑移向量判斷臨界分解剪應 力與尺寸關係圖........................................................................113 表目錄 表 2.1 預測修正法之修正參數..........................................................31 表2.2 各種晶體結構之雙晶平面與雙晶方向..................................31 表3.1 Finnis-Sinclair勢能函數之參數...............................................56 表3.2   Fe塊材之彈性係數與文獻對照表..................................56 表3.3 [100]奈米線的各組滑移系統之施密特因子.......................57 表3.4 [110]奈米線的各組滑移系統之施密特因子........................60 表3.5 [111]奈米線的各組滑移系統之施密特因子...........................63 表3.6 滑移向量法判斷奈米線之施密特因子..................................66 表4.1 Ackland勢能函數之參數..........................................................95 表4.2 塊材鈦彈性係數與文獻對照表..............................................96 表4.3 [0001]、[2-1-10]、[01-10]奈米線的各組滑移系統之施密特因子 .......................................................................................................97 表4.4 滑移向量法判斷奈米線之施密特因子................................100 表4.5 [10-10]、[10-11]奈米線之施密特因子...................................100

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