近年來有許多學者都不斷地鑽研奈米尺度下之材料性質，單一金 屬結構於奈米尺度下，是否會因為尺寸效應與表面效應而改變材料的 楊氏係數。本研究主要在探金奈米線受到單軸拉伸時，其應力應變關 係討論，以及其塑性變形的產生。
以分子動力學來模擬巨觀的拉伸試驗，在微觀世界中奈米線會存 在一壓縮預應力，與巨觀拉伸試驗一開始近乎為零有很大的不同。在 應力應變曲線中，分子動力學模擬出來的結果並沒有像巨觀應力曲線 那樣平滑，會有微小應力震盪的現象，是因為拉伸過程中原子各別在 相對位置作平衡所以造成了曲線的微小震盪。在彈性區間內，原子雖 然有小幅的移動，不過每顆原子與其相鄰原子的相對位置並不會改 變，在降伏點時，此時原子相對位置還未改變，原子間勢能達到極值，
下一時間點原子瞬間滑移成核產生差排，應力值達最高後開始下降，
而其滑移平面為{111}面心立方的最緊密堆積平面。及其滑移平面不 斷重覆差排、滑動、形成另一個穩定結構的過程，此為降伏後應力應 變關係成鋸齒狀的主要原因。進一步改變模擬的奈米線尺寸後，降伏 應力會隨尺寸變大而變小，而主要原因為尺寸變大時，奈米線的表面 積因此而變大，使得差排成核的機率大大增加，而楊氏係數在[100]
40
方向會隨尺寸變大而有增加的趨勢，原因為其表面積對總體積的比值 會隨尺寸變大而減少，當減至趨近於零時，此時材料的表面效應對結 構影響已經非常小，材料性質接近塊材。拉伸應變率的部份，分子模 擬的結果與巨觀實驗的結果有相同的趨勢，即楊氏係數不會隨應變率 變化而改變，它會在一定值上下跳動。應變率對降伏強度有較顯著的 影響，因為滑移面的生成需要一段時間，應變率太大時，滑移面無法 生成以致於降伏強度不斷增加。此外，奈米線的強度及塑性變形的模 式，會因原子的排列(方向性)而有很大的差異。
在討論能量損耗的部份，雖然尺寸越大的奈米線其降伏後能量損 耗會越大，但如果我們將能量去除上系統體積後，發現在不同尺寸下 單位體積的能量差別不大，說明奈米線單位體積的滑移程度是與尺寸 無關的。在方向[111]的奈米線能量損耗明顯會比[100]方向來得小，
因為在[111]方向幾乎沒有滑移面生成。
奈米線含缺陷的模型討論空孔率對其結構強度的影響，發現空孔 率越大的奈米線，其降伏強度也會明顯地下降，但對結構的楊氏係數 幾乎沒影響，在小缺陷率(約1%)以下的情況，材料在降伏前仍維持線 彈性。另一方面比較缺陷位置的影響性，發現相同缺陷率下含構槽的 奈米線強度比含空孔奈米線小很多，因含構槽奈米線有更強的應力集 中效應。此外可以觀察得到奈米線的差排成核位置與缺陷位置有很密
41
切的關係，幾乎都會從缺陷附近最先開始發生差排成核的現象。而能 量損耗則以缺陷率最大的模型最小，即空孔越大其滑移程度規模最 小。
42
參考文獻
[1] W. Haiss, 2001, Surface Stress of Clean and Adsorbate-Covered Solids, Report on Progress in Physics, Vol. 64, pp. 591-648.
[2] Y. Gao, F. Wang, T. Zhu and J. Zhao, 2010, Investigation on the Mechanical Behaviors of Copper Nanowires under Torsion,
Computational Materials Science , Vol. 49, pp. 826-830.
[3] L. Sandoval and H. M. Urbassek, 2009, Solid–Solid Phase Transitions in Fe Nanowires Induced by Axial Strain,
Nanotechnology, Vol. 20, p. 325704.
[4] Y. H. Wen, Z. Z. Zhu and R. Z. Zhu, 2008, Molecular Dynamics of Study of the Mechanical Behavior of Nickel Nanowire: Strain Rate Effects, Computational Materials Science, Vol. 41, pp. 553-560.
[5] D. L. Chen and T. C. Chen, 2005, Mechanical Properties of Au Nanowires under Uniaxial Tension with High Strain-Rate by Molecular Dynamics, Nanotechnology, Vol. 16, pp. 2972-2981.
[6] H. S. Park and J. A. Zimmerman, 2005, Modeling Inelasticity and Failure in Gold Nanowires, Physical Review B, Vol. 72, p. 054106.
[7] A. R. Setoodeh, H. Atariani and M. Khosrownejad, 2008, Nickel Nanowires under Uniaxial Loads: A Molecular Dynamics Simulation Study, Computational Materials Science, Vol. 44, pp.
378-383.
[8] H. A. Wu, 2006, Molecular Dynamics Study of the Mechanics of Metal Nanowires at Finite Temperature, European Journal of
Mechanics A/Solids, Vol. 25, pp. 370-377.
[9] Y. Gan and J. K. Chen, 2009, Molecular Dynamics Study of Size, Temperature and Strain Rate Effects on Mechanical Properties of Gold Nanofilms, Applied Physics A, Vol. 95, pp. 357-362.
43
[10] W. Liang and M. Zhou, 2003, Size and Strain Rate Effects in Tensile Deformation of Cu Nanowire, Nanotechnology, Vol. 2, pp. 324-327.
[11] H. S. Park, K. Gall and J. A. Zimmerman, 2006, Deformation of FCC Nanowires by Twinning and Slip, Journal of the Mechanics
and Physics of Solids, Vol.54, pp. 1862-1881.
[12] H. A. Wu, 2006, Molecular Dynamics Study on Mechanics of Metal Nanowire, Mechanics Research Communications, Vol. 33, pp. 9-16.
[13] J. Diao, K. Gall, M. L. Dunn and J. A. Zimmerman, 2006, Atomic Simulations of the Yielding of Gold Nanowires, Acta Materialia, Vol.
54, pp. 643-653.
[14] C. Ji and H. S. Park, 2006, Geometric Effects on the Inelastic Deformation of Metal Nanowires, Applied Physics Letters, Vol. 89, p.
181916.
[15] D. Huang, P. Qiao and F. Asce, 2011, Mechanical Behavior and Size Sensitivity of Nanocrystalline Nickel Wires Using Molecular Dynamics Simulation, Journal of Aerospace Engineering, Vol.
24, pp. 147-153.
[16] D. Huang, Q. Zhang and P. Qiao, 2011, Molecular Dynamics Evaluation of Strain Rate and Size Effects on Mechanical Properties of FCC Nickel Nanowires, Computational Materials Science, Vol.
50, pp. 903-910.
[17] D. Wang, J. Zhao, S. Hu, X. Yin, S. Liang, Y. Liu and S. Deng, 2007, Where, and How, Does a Nanowire Break?, Nano Letters, Vol.
7, pp. 1208-1212.
[18] Y. Liu, J. Zhao and F. Wang, 2009, Influence of Length on Shock-Induced Breaking Behavior of Copper Nanowires, Physical
Review B, Vol. 80, p. 115417.
[19] F. Wang, Y. Gao, T. Zhu and J. Zhao, 2011, Shock-Induced Breaking in the Gold Nanowire with the Influence of Defects and
44
Strain Rates, Nanoscale, Vol. 3, pp. 1624-1631.
[20] S. Y. Lin, T. H. Wang, T. H. Fang and W. L. Chen, 2010, Effects of Pores and Temperature on Mechanics of Gold Nanowires Using Molecular Dynamics, Current Applied Physics, Vol. 10, pp. 5-9.
[21] Y. Jia, H. Liu and L. Yu, 2009, Propagation of Stress Wave in Nickel Single Crystals Nanofilm with Cavity Via Molecular Dynamics, Proceedings of the IEEE International Conference on
Automation and Logistics Shenyang, pp. 1401-1405.
[22] H. A. Wu, G. R. Liu and J. S. Wang, 2004, Atomistic and continuum Simulation on Extension Behaviour of Single Crystal with Nano-Holes, Modeling and Simulation in Material Science and
Engineering, Vol. 12, pp. 225-233.
[23] P. Heino, H. Hakkinen and K. Kaski, 1998, Molecular-Dynamics Study of Copper with Defects under Strain, Physical Review B, Vol.
58, pp. 641-652.
[24] X. Yang, T. Zhou and C. Chen, 2007, Effective Elastic Modulus and Atomic Stress Concentration of Single Crystal Nano-Plate with Void,
Computational Materials Science, Vol. 40, pp. 51-56.
[25] R. S. Mcentire and Y. L Shen, 2006, An Atomics Analysis of Incipient Metal Plasticity During Tensile Loading, Molecular
Simulation, Vol. 32, pp. 857-867.
[26] Q. K. Li and M. Li, 2006, Molecular Dynamics Simulation of Intrinsic and Extrinsic Mechanical Properties of Amorphous Metals,
Intermetallics, Vol. 14, pp. 1005-1010.
[27] Y. G. Xu, K. Behdinan and Z. Fawaz, 2004, Molecular Dynamics Calculation of the J-Integral Fracture Criterion for Nano-Sized Crystals, International Journal of Fracture, Vol. 130, pp. 571-582.
[28] Y. G. Xu, K. Behdinan and Z. Fawaz, 2004, Stepwise Equilibrium and Adaptive Molecular Dynamics Simulation for Fracture
45
Toughness of Single Crystals, Journal of Intelligent Material
Systems and Structures, Vol.15, pp. 933-939.
[29] M. Karimi, T. Roarty and T. Kaplan, 2006, Molecular Dynamics Simulations of Crack Propagation in Ni with Defects, Modeling
Simulation Material Science Engineering, Vol. 14, pp. 1409-1420.
[30] H. Zhou and S. Qu, 2009, The Effect of Nanoscale Twin Boundaries on Fracture Toughness in Nanocrystalline Ni, Nanotechnology, Vol.
21, p. 035706.
[31] K. N. Solanki, D. K. Ward and D. J. Bannmann, 2011, A Nanoscale Study of Dislocation Nucleation at the Crack Tip in the Nickel-Hydrogen System, Metallurgical and materials transactions
A, Vol. 42, pp. 340-347.
[32] Q. Pu, Y. Leng, L. Tesetseris, H. S. Park, S. T. Pantelides and P. T.
Cummings, 2007, Molecular Dynamics Simulations of Stretched Gold Nanowires: The Relative Utility of Different Semiempirical Potentials, The Journal of Chemical Physics, Vol. 126, p. 144707.
[33] S. J. Plimpton, 1995, Fast Parallel Algorithms for Short Range Molecular Dynamics, Journal of Computation Physics, Vol. 117, pp.
1-19.
[34] S. M. Foiles, M. I. Baskes and M. S. Daw, 1986, Embedded-Atom-Method- Functions for the Fcc Metals Cu, Ag, Au, Ni, Pd, Pt, and their Alloys, Physical Review B, Vol. 33, pp.7983-7991.
[35] J. E. Lennard-Jones, 1931, Cohesion, Proceedings of the Physical
Society, Vol. 43, pp. 461-482.
[36] I. G. Kaplan, 2003, Handbook of Molecular Physics and Quantum
Chemistry, Wiley.
[37] T. C. Choy, 1999, Effective Medium Theory: Principles and
Applications, Oxford Press.
46
[38] M. Z. Bazant and E. Kaxiras, 1997, Environment Dependent Interatomic Potential for Bulk Silicon, Physics Review B, Vol. 56, p.
85427.
[39] A. P. Sutton and J. Chen, 1990, Long-Range Finnis-Sinclair Potentials, Philosophical Magazine Letter, Vol. 61, pp. 139-146.
[40] B. J. Alder and T. E. Wainwright, 1959, Studies in Molecular Dynamics. I. General Method, Journal of Chemical Physics, Vol. 31, pp. 459-466.
[41] J. A. Zimmerman, E. B. Webb, J. J. Hoyt, R. E. Jones, P. A. Klein and D. J. Bammann, 2004, Calculation of Stress in Atomistic Simulation, Modeling Simulation Material Science Engineering, Vol.
12, pp. 319-332.
[42] Z. S. Basinski, M.S. Duesbery and R. Taylor, 1971, Influence of Shear Stress on Screw Dislocations in a Model Sodium Lattice,
American Journal of Physics, Vol. 49, pp. 2160-2180.
[43] N. Miyazaki and Y. Shiozaki, 1996, Calculation of Mechanical Properties of Solids Using Molecular Dynamics Method,
International Journal Series A-Mechanics and Material Engineering,
Vol. 39, pp. 606-612.[44] Y. C. Lin and D. J. Pen, 2007, Atomic Behavior Analysis of Cu Nanowire Under Uniaxial Tension with Maximum Local Stress Method, Molecular Simulation, Vol. 33, pp.979-988.
[45] C. L. Kelchner, S. J. Plimpton and J. C. Hamilton, 1998, Dislocation Nucleation and Defect Structure During Surface Indentation,
Physical Review B, Vol. 58, pp. 11085-11088.
[46] J. Diao, K. Gall and M. L. Dunn, 2003, Surface Stress Induced Phase Transformation in Metal Nanowires, Nature Materials, Vol. 2, pp. 656-660.
[47] J. P. Hirth and J. Lothe, 1982, Theory of Dislocations, Wiley, New
47
York.
[48] 王忠, 機械工程材料, 2005, 北京清華大學出版社.
[49] 王純敏,考慮應變速率的強度分析, 2010, 上海科技大學.
[50] 章荣建, 機電技術, 2008, 南平出版社, 第二期出版.
[51] W. D. Callister, 1994, Materials Science and Engineering, Wiley, New York.
[52] J. Gaspar, M. E. Schmidt, J. Held and O. Paul, Evaluation of the Mechanical Properties of Aluminum Thin Films as a Function of Strain Rate Using the Wafer-Scale Microtensile Technique, 2009,
Materials Research Society, Vol. 1139.
[53] K. Jonnalagadda, N. Karanjaokar, I. Chasiotis, J. Chee and D.
Peroulis, 2010, Strain Rate Sensitivity of Nanocrystalline Au Films at Room Temperature, Acta Materialia, Vol. 58, pp. 4674-4684.
[54] Y. Umeno, T. Shimada and T. Kitamura, 2010, Dislocation Nucleation in a Thin Cu Film from Molecular Dynamics Simulations: Instability Activation by Thermal Fluctuations,
Physical Review B, Vol, 82, pp. 104-108.
[55] B. Wu, A. Heidelberg and J. J. Boland, 2005, Mechanical Properties of Ultrahigh-Strength Gold Nanowire, Nature Materials, Vol. 4, pp.
525-529.
48
附 表
表 1 模擬材料基本設定
晶格常數(a) 4.08 Å
原子模型 FCC結構
系統溫度 1.0 K
變形條件 等位移拉伸變形
表 2 模擬材料參數表[34]
Z
0 ns
11.0 1.4475 0.1269 2.0 1.0809
表 3 金塊材勁度矩陣參數
C11 C22 C33 C12 C13 C23
174.2 174.2 174.2 136.9 136.9 136.9
單位:GPa
49
表 4 不同尺寸之應力應變數據
Size 8a 12a 16a 20a
Yield stress(GPa) 4.46 4.28 4.19 4.16
Yield strain 0.093 0.085 0.080 0.079
Young’s modulus(GPa) 47.20 49.90 51.50 51.77 a:晶格常數
表 5 應變率計算
Strain Rate(s-1) 2x10-7 2x10-8 2x10-9
Strain/Step 0.1% 0.1% 0.1%
Equilibrium Time (ps) 50 5 0.5
表 6 三晶向楊氏係數
Orientation [100] [110] [111] Exp.[55]
Yield stress(GPa) 4.46 3.67 3.18 3.5-5.6
Young’s modulus(GPa) 47.20 89.18 106.81 7011
50
表 7 不同尺寸臨界應變平衡前後能量差異
Size 8a 12a 16a 20a
Energy loss(eV) 118.6 260.6 453.5 732.1
Energy loss/vol 6.82E-4 6.66E-4 6.52E-4 6.73E-4
表 8 不同晶向臨界應變平衡前後能量差異
Orientation [100] [110] [111]
Energy loss(eV) 118.6 61.9 20.1
Energy loss/vol 6.82E-4 3.56E-4 1.12E-4
表 9 缺陷率計算
State Vacancies Disorder %
perfect 0 0.0
2x3 51 0.44
2x5 85 0.73
2x7 119 1.02
51
表 10 不同缺陷率臨界應變平衡前後能量差異
Energy perfect 2x3 2x5 2x7
Energy loss(eV) 118.6 108.3 89.5 68.4
Energy loss/vol 6.82E-4 6.27E-4 5.22E-4 4.01E-4
表 11 不同缺陷位置臨界應變平衡前後能量差異
Energy perfect 2x3 vacancy 2x3 notch
Energy loss(eV) 118.6 108.3 52.89
Energy loss/vol 6.82x10-4 6.27x10-4 3.06x10-4
表 12 FCC 滑移系統
Plane (111) (-1 -1 1)
Direction [0 -1 1] [1 0 -1] [-1 1 0] [0 1 1] [-1 0 1] [1 -1 0]
System a1 a2 a3 b1 b2 b3
Plane (-111) (1 -1 1)
Direction [0 -1 1] [-1 0 -1] [1 1 0] [0 1 1] [1 0 -1] [-1 -1 0]
System c1 c2 c3 d1 d2 d3
52
表 13 [100]晶向拉伸之 schmid’s factor
System a1 a2 a3 b1 b2 b3
m 0 0.408 0.408 0 0.408 0.408
System c1 c2 c3 d1 d2 d3
m 0.408 0.408 0.408 0 0.408 0.408
表 14 [110]晶向拉伸之 schmid’s factor
System a1 a2 a3 b1 b2 b3
m 0.408 0.408 0 0.408 0.408 0
System c1 c2 c3 d1 d2 d3
m 0 0 0 0 0 0
表 15 [111]晶向拉伸之 schmid’s factor
System a1 a2 a3 b1 b2 b3
m 0 0 0 0.272 0 0
System c1 c2 c3 d1 d2 d3
m 0 0 0.272 0.272 0 0
53
附 圖
圖 2.1 截斷半徑 rc示意圖
圖 2.2 Centro-Symmetry Parameter 計算之原子示意圖
54
圖 2.3 Schmid’s Law 示意圖
X
Y Z
圖 3.1 金奈米線結構
55
圖 3.2 拉伸示意圖
X Y
Z
圖 3.3 金塊材示意圖
56
57 Time (ps)
Stress(GPa)
0 25 50 75 100
-2 -1 0 1 2 3 4
XX
YY
ZZ
圖 3.6 8a x 8a 結構應力對時間變化
Time (ps)
Energy(eV)
0 25 50 75 100
-44600 -44550 -44500
圖 3.7 8a x 8a 結構能量對時間變化
58
圖 3.8 6a x 6a 平衡後結構圖
圖 3.9 8a x 8a 平衡後結構圖
圖 3.10 12a x 12a 平衡後結構圖
圖 3.11 16a x 16a 平衡後結構圖
59
Time (ps)
Equilibriumstrain%
0 25 50 75 100
-20 -15 -10 -5 0 5 10 15
6a x 6a 8a x 8a 12a x 12a 16a x 16a 20a x 20a
圖 3.12 平衡應變量對時間關係
圖 3.13 不同截面下應力應變關係圖(strain rate=2x107s-1)
60
Cubic lattice unit (a)
Young'sModulus(GPa)
8 10 12 14 16 18 20
35 40 45 50 55 60 65
bulk
圖 3.14 不同截面與楊氏係數關係(strain rate=2x107s-1)
Cubic lattice unit (a)
Yieldstress(GPa)
8 10 12 14 16 18 20
4 5
圖 3.15 不同截面與降伏應力關係(strain rate=2x107s-1)
61
圖 3.16 金奈米線結構在應變率為 2x10-7s-1下變形圖 ([100]晶 向)
62
圖 3.17 降伏點(strain=0.093) 時其應力變化(a)t=10ps(b)t=20ps (c)t=30ps (d)t=40ps (e)t=50ps
63
圖 3.18 CSP 分佈圖(a)ε=0.092 (b)ε=0.093 t=10ps (c)ε=0.093
t=20ps (d)ε=0.093 t=30ps (e) ε=0.093 t=40ps (f)ε=0.093 t=50ps
64
圖 3.19 csp 分佈圖 ε=0.093 (a)~(e)分別代 t=18~t=22ps (f) t=35ps (g) t=40ps (h) t=45ps (i) t=50ps (j) t=80ps
圖 3.20 鋸齒狀應力示意圖
65
圖 3.21 應變率分別為 2 x107 S-1、2 x108 S-1、2 x109 S-1應力應 變關係圖
Strain rate (s-1)
Yieldstress(GPa)
107 108 109
4 4.4 4.8 5.2
log
圖 3.22 應變率對降伏應力關係
66 Strain rate (s-1)
Yieldstrain
107 108 109
0.08 0.09 0.1 0.11 0.12
log
圖 3.23 應變率對降伏應變關係
67
圖 3.24 [100]、[110]及[111]晶向
68
圖 3.25 8 倍晶格邊長在應變率 2 x107s-1下應力應變圖( [110]
晶向)
69
(a)
strain=0.025
(b)
strain=0.030 t=10ps
(c)
strain=0.030 t=20ps
(d)
strain=0.030 t=30ps
圖 3.26 金奈米線結構變形圖( [110]晶向)
70 (b)strain=0.03,t=10ps (c) strain=0.03,t=20ps (d) strain=0.03,t=30ps
71
圖 3.28 8 倍晶格邊長在應變率 2 x107s-1下應力應變圖([111]晶 向)
72
圖 3.29 金奈米線結構變形圖([111]晶向)
73
圖 3.30 [111] 晶向拉伸變形 CSP 圖 (a)strain=0.027 (b)strain=0.031 (c) strain=0.032 (d) strain=0.033
74
圖 3.31 [100]、[110] 、[111]晶向拉伸應力應變圖
圖 3.32 坐標軸轉換
75
圖 3.33 奈米線差排成核點 (a) [100] (b) [110]
Time (ps)
Energy(eV)
0 25 50 75 100
-44540 -44520 -44500 -44480 -44460 -44440 -44420
圖 3.34 8 倍晶格常數線寛在臨界應變下勢能變化圖
76
[100] [110] [111]
圖 3.36 降伏應力及能量損耗對方向性關係
77
78
圖 4.2 2x3、2x5 及 2x7 三種空孔率示意圖
圖 4.3 穩定前後含空孔結構(a) 2x3 (b) 2x5 (c) 2x7
79
圖 4.4 含空孔奈米線應力應變圖
Disorder %
Yieldstress(GPa)
0 0.25 0.5 0.75 1
0 1 2 3 4 5 6 7
圖 4.5 奈米線缺陷率對降伏應力關係圖
80
圖 4.6 2x3 空孔奈米線在不同應變下結構圖
81
圖 4.7 含 2X3 空孔結構臨界應變 ε=0.087 下 CSP 分佈圖(a)t=10 (b)t=14 (c)t=15 (d)t=16 (e)t=17 (f)t=18 (g)t=20 (h)t=30 (i)t=40
(j)t=50
82
perfect 2x3 2x5 2x7
圖 4.8 降伏應力及能量損耗對缺陷率的關係
圖 4.9 含溝槽奈米線示意
83
圖 4.10 含溝槽奈米線結構穩定
圖 4.11 含構槽奈米線拉伸變形圖
84
圖 4.12 含溝槽 CSP 分佈圖 ε=0.068 (a)t=10ps(b)t=20 ps (c) t=30ps (d) t=40ps (e) t=50ps
85
perfect 2x3 vacancy 2x3 notch
圖 4.14 降伏應力及能量損耗對缺陷型式的關係
86
圖 4.15 含 2x3 空孔及溝槽應力應變圖
圖 4.16 缺陷位置應力集中程度比較 (a)含溝構缺陷 (b)含空孔 缺陷
87
圖 4.17 奈米線差排成核點 (a) 2x3 notch (b) 2x3 vacancy (c) 2x5 vacancy (d) 2x7 vacancy