臺灣博碩士論文加值系統

本研究透過結合密度泛函理論（DFT），粒子群優化（PSO），最大熵方法和分子動力學（MD）模擬來開發設計新型高熵合金（HEA）的完整模擬程序。採用PSO擬合的參數的第二近鄰修正嵌入原子法（2NN MEAM）勢將用於MD方法。 通過該程序，將預測不同元素組成的最穩定的HEA結構。通過利用MD模擬的優點，可以預測詳細的HEA結構及其機械性能，包括楊氏模數、體積模數和強度。拉伸模擬將系統地應用於具有不同元素組成的HEA，用於觀察HEA斷裂機理並理解機械性質與HEA元素組成之間的關係。
從應力應變曲線可求得楊氏模數約為146.28 GPa，與實驗值122 GPa 十分的相近，且能預測不同比例元素的材料性質趨勢。但局部分析高熵合金材料受應力變形時，因勢能參數的缺陷使只能準確的預測AlMo0.5NbTa0.5TiZr HEA 線性彈性階段的表現，但無法正確的預測滑移、成核與破壞的形成，我們推測原因在於，擬合參數時過少的滑移數據作為參考結構，導致 HEA 模擬拉伸試驗到屈服階段後的表現不佳，最後分析顯示出AlMo0.5NbTa0.5TiZr其六種元素的電荷分佈，進而影響電子交換的情形與抗腐蝕性質。

This study combined density functional theory (DFT), particle swarm optimization (PSO), maximum entropy method and molecular dynamics (MD) simulation to develop and design a complete simulation program for new high entropy alloys (HEA). The second nearest neighbor modified embedding atomic method (2NN MEAM) is applied to MD method. Through this program, the most stable HEA structure composed of different elements is predicted. Detailed HEA structures and mechanical properties, including young''s modulus, bulk modulus and strength, can be predicted by using the advantages of MD simulation. Tensile simulation is applied systematically to HEA with different elements to observe the fracture mechanism of HEA and understand the relationship between mechanical properties and HEA element composition.
According to the stress-strain curve, young''s modulus is about 146.28 GPa, which is very close to the experimental value of 122 GPa, and it can predict the trend of material properties of elements with different proportions. But local analysis high entropy alloys by the deformation and stress, because of the defects of potential energy parameters that can accurately predict AlMo0.5 NbTa0.5 TiZr HEA linear elastic stage performance, but not the right to predict the formation of the slip, nucleation and destruction, we speculated that the reason is that too little fitting parameters of sliding data as a reference structure, leading to HEA simulate tensile test to yield stage after poor performance, in the final analysis shows AlMo0.5 NbTa0.5 TiZr charge distribution of the six elements, affecting the situation of the electronic exchange and corrosion resistant properties.

論文審定書 i
誌謝 ii
中文摘要 iii
Abstract iv
目錄 vi
圖目錄 xi
表目錄 xiii
1. 緒論 1
1.1 研究動機與目的 1
1.2 高熵合金文獻回顧 3
1.3 論文架構 7
2. 理論基礎及方法 8
2.1 密度泛函理論(Density Functional Theory) 8
2.1.1 電子密度 8
2.1.2 湯瑪士-費米理論 (Thomas-Fermi Theory) 9
2.1.3 霍恩貝格-科恩理論 (Hohenberg-Kohn Theory) 9
2.1.4 科恩-沈方程式 (Kohn–Sham equation) 10
2.1.5 交換-相關函數(Exchange-Correlation Function) 11
2.2 勢能函數 12
2.2.1 原子間作用勢能 12
2.2.2 Particle Swarm Optimization (PSO) 14
2.3 最大熵值法理論 (Maximum Entropy (MaxEnt) methed) 15
2.4 分子靜力學理論 19
2.4.1 L-BFGS演算法 19
2.4.2 火炎演算法 21
2.4.3 共軛梯度法 22
2.5 分子動力學理論 24
2.5.1 積分法則 24
2.5.2 諾斯-胡佛恆溫法(Nosé-Hoover thermostat) 25
2.5.3 時間步階選取 26
2.6 結構分析 28
2.6.1 原子級應力分析 28
2.6.2 區域應變分析 30
2.7 週期邊界的處理 32
2.8 鄰近原子表列數值方法 33
2.8.1 截斷半徑法 33
2.8.2 Verlet List表列法 34
2.8.3 Cell Link表列法 36
2.8.4 Verlet List結合Cell Link表列法 37
2.9 模擬流程 38
3. 結果與討論 41
3.1 擬合勢能參數 42
3.1.1 尋找最佳DFT設定及建立參考資料 42
3.1.2 參數擬合結果 45
3.2 模型建立及分析 55
3.2.1 試驗之物理模型建構 55
3.2.2 試驗之物理模型分析 57
3.3 拉伸試驗與機械性質探討 59
3.3.1 拉伸模型建立 59
3.3.2 試驗結果分析 60
3.4 電子與腐蝕性質探討 64
3.4.1 電化學性質分析模型建立 64
3.4.2 電子性質分析 65
3.5 結論 68
3.6 建議與未來展望 70
參考文獻 71

[1]H. Daraee et al., "Application of gold nanoparticles in biomedical and drug delivery," vol. 44, no. 1, pp. 410-422, 2016.
[2]P. J. G. b. Goodman, "Current and future uses of gold in electronics," vol. 35, no. 1, pp. 21-26, 2002.
[3]G. Cole and A. J. M. c. Sherman, "Light weight materials for automotive applications," vol. 35, no. 1, pp. 3-9, 1995.
[4]M. Nakai, T. J. M. S. Eto, and E. A, "New aspect of development of high strength aluminum alloys for aerospace applications," vol. 285, no. 1-2, pp. 62-68, 2000.
[5]O.-S. Kwon and A. J. E. s. Elnashai, "The effect of material and ground motion uncertainty on the seismic vulnerability curves of RC structure," vol. 28, no. 2, pp. 289-303, 2006.
[6]C. F. Pare, Metals make the world go round: the supply and circulation of metals in Bronze Age Europe: proceedings of a conference held at the University of Birmingham in June 1997. Oxbow Books Limited, 2000.
[7]R. E. J. J. Sanders, "Technology innovation in aluminum products," vol. 53, no. 2, pp. 21-25, 2001.
[8]N. Tsuji, R. Ueji, Y. Minamino, and Y. J. S. M. Saito, "A new and simple process to obtain nano-structured bulk low-carbon steel with superior mechanical property," vol. 46, no. 4, pp. 305-310, 2002.
[9]P. L. Mangonon and G. J. M. t. Thomas, "The martensite phases in 304 stainless steel," vol. 1, no. 6, pp. 1577-1586, 1970.
[10]J.-W. J. A. d. C. S. d. M. Yeh, "Recent progress in high-entropy alloys," vol. 31, no. 6, pp. 633-648, 2006.
[11]O. Senkov, C. Woodward, and D. J. J. Miracle, "Microstructure and properties of aluminum-containing refractory high-entropy alloys," vol. 66, no. 10, pp. 2030-2042, 2014.
[12]Y. Zhang et al., "Microstructures and properties of high-entropy alloys," vol. 61, pp. 1-93, 2014.
[13]L. Xie, P. Brault, A.-L. Thomann, X. Yang, Y. Zhang, and G. J. I. Shang, "Molecular dynamics simulation of Al–Co–Cr–Cu–Fe–Ni high entropy alloy thin film growth," vol. 68, pp. 78-86, 2016.
[14]L. Xie, P. Brault, A.-L. Thomann, and J.-M. J. A. S. S. Bauchire, "AlCoCrCuFeNi high entropy alloy cluster growth and annealing on silicon: A classical molecular dynamics simulation study," vol. 285, pp. 810-816, 2013.
[15]J. C. J. S. Huang, "Evaluation of Tribological Behavior of Al‐Co‐Cr‐Fe‐Ni High Entropy Alloy Using Molecular Dynamics Simulation," vol. 34, no. 5, pp. 325-331, 2012.
[16]J. Li, Q. Fang, B. Liu, Y. Liu, and Y. J. R. A. Liu, "Mechanical behaviors of AlCrFeCuNi high-entropy alloys under uniaxial tension via molecular dynamics simulation," vol. 6, no. 80, pp. 76409-76419, 2016.
[17]P. Hohenberg and W. Kohn, "Inhomogeneous electron gas," Physical review, vol. 136, no. 3B, p. B864, 1964.
[18]W. Kohn and L. J. Sham, "Self-consistent equations including exchange and correlation effects," Physical Review, vol. 140, no. 4A, p. A1133, 1965.
[19]J. C. Slater, "A simplification of the Hartree-Fock method," Physical Review, vol. 81, no. 3, p. 385, 1951.
[20]M. P. Allen and D. J. Tildesley, Computer simulation of liquids. Oxford university press, 1989.
[21]P. Hohenberg and W. Kohn, "INHOMOGENEOUS ELECTRON GAS," (in English), Physical Review B, Article vol. 136, no. 3B, pp. B864-&, 1964.
[22]W. Kohn and L. J. Sham, "Self-Consistent Equations Including Exchange and Correlation Effects," (in English), Physical Review, vol. 140, no. 4A, pp. 1133-&, 1965.
[23]N. Andrei, "Scaled memoryless BFGS preconditioned conjugate gradient algorithm for unconstrained optimization," (in English), Optimization Methods & Software, vol. 22, no. 4, pp. 561-571, Aug 2007.
[24]J. Han et al., "Tailoring the degradation and biological response of a magnesium–strontium alloy for potential bone substitute application," Materials Science and Engineering: C, vol. 58, pp. 799-811, 2016.
[25]E. A. Brandes and G. Brook, Smithells metals reference book. Elsevier, 2013.
[26]N. Chandra, S. Namilae, and C. Shet, "Local elastic properties of carbon nanotubes in the presence of Stone-Wales defects," (in English), Physical Review B, vol. 69, no. 9, Mar 2004.
[27]M. J. R. e. Karolewski and d. i. solids, "Tight-binding potentials for sputtering simulations with fcc and bcc metals," vol. 153, no. 3, pp. 239-255, 2001.
[28]H. C. Hsu, J. H. Chien, J. S. Huang, L. M. Chu, and S. L. Fu, "Nanoscale bondability between Cu-Al intermetallic compound for Cu wirebonding," in Electronics Packaging (ICEP), 2014 International Conference on, 2014, pp. 618-622.
[29]M. Baskes, "Modified embedded-atom potentials for cubic materials and impurities," Physical Review B, vol. 46, no. 5, p. 2727, 1992.
[30]A. J. Cao, Y. Q. Cheng, and E. Ma, "Structural processes that initiate shear localization in metallic glass," (in English), Acta Materialia, vol. 57, no. 17, pp. 5146-5155, Oct 2009.
[31]J. Kennedy, "Particle swarm optimization," in Encyclopedia of machine learning: Springer, 2011, pp. 760-766.
[32]W. W. Hager and H. Zhang, "A new conjugate gradient method with guaranteed descent and an efficient line search," SIAM Journal on optimization, vol. 16, no. 1, pp. 170-192, 2005.
[33]L. Zhang, W. Zhou, and D. Li, "Some descent three-term conjugate gradient methods and their global convergence," Optimisation Methods and Software, vol. 22, no. 4, pp. 697-711, 2007.
[34]W. Quapp, "A growing string method for the reaction pathway defined by a Newton trajectory," The Journal of chemical physics, vol. 122, no. 17, p. 174106, 2005.
[35]H. Bruck, S. McNeill, M. A. Sutton, and W. Peters, "Digital image correlation using Newton-Raphson method of partial differential correction," Experimental mechanics, vol. 29, no. 3, pp. 261-267, 1989.
[36]C. Broyden, "The convergence of single-rank quasi-Newton methods," Mathematics of Computation, vol. 24, no. 110, pp. 365-382, 1970.
[37]R. Fletcher, "A new approach to variable metric algorithms," The computer journal, vol. 13, no. 3, pp. 317-322, 1970.
[38]D. Goldfarb, "A family of variable-metric methods derived by variational means," Mathematics of computation, vol. 24, no. 109, pp. 23-26, 1970.
[39]D. F. Shanno, "Conditioning of quasi-Newton methods for function minimization," Mathematics of computation, vol. 24, no. 111, pp. 647-656, 1970.
[40]E. Bitzek, P. Koskinen, F. Gähler, M. Moseler, and P. Gumbsch, "Structural relaxation made simple," Physical review letters, vol. 97, no. 17, p. 170201, 2006.
[41]D. C. Liu and J. Nocedal, "On the limited memory BFGS method for large scale optimization," Mathematical programming, vol. 45, no. 1-3, pp. 503-528, 1989.
[42]O. Teleman, B. Jönsson, and S. Engström, "A molecular dynamics simulation of a water model with intramolecular degrees of freedom," Molecular Physics, vol. 60, no. 1, pp. 193-203, 1987.
[43]S. Nosé, "A unified formulation of the constant temperature molecular dynamics methods," The Journal of chemical physics, vol. 81, no. 1, pp. 511-519, 1984.
[44]M.-S. Lee, S. Chacko, and D. Kanhere, "First-principles investigation of finite-temperature behavior in small sodium clusters," The Journal of chemical physics, vol. 123, no. 16, p. 164310, 2005.
[45]S. M. Ghazi, M.-S. Lee, and D. Kanhere, "The effects of electronic structure and charged state on thermodynamic properties: An ab initio molecular dynamics investigations on neutral and charged clusters of Na 39, Na 40, and Na 41," The Journal of chemical physics, vol. 128, no. 10, p. 104701, 2008.
[46]N. Chandra, S. Namilae, and C. Shet, "Local elastic properties of carbon nanotubes in the presence of Stone-Wales defects," Physical Review B, vol. 69, no. 9, p. 094101, 2004.
[47]N. Tokita, M. Hirabayashi, C. Azuma, and T. Dotera, "Voronoi space division of a polymer: Topological effects, free volume, and surface end segregation," The Journal of chemical physics, vol. 120, no. 1, pp. 496-505, 2004.
[48]H. Hsu, J. Chien, J. Huang, L. Chu, and S. Fu, "Nanoscale bondability between Cu-Al intermetallic compound for Cu wirebonding," in Electronics Packaging (ICEP), 2014 International Conference on, 2014, pp. 618-622: IEEE.
[49]D. Srolovitz, K. Maeda, V. Vitek, and T. Egami, "Structural defects in amorphous solids statistical analysis of a computer model," Philosophical Magazine A, vol. 44, no. 4, pp. 847-866, 1981.
[50]D. Srolovitz, K. Maeda, V. Vitek, and T. Egami, "Structural Defects in Amorphous Solids Statistical-Analysis of a Computer-Model," (in English), Philosophical Magazine a-Physics of Condensed Matter Structure Defects and Mechanical Properties, vol. 44, no. 4, pp. 847-866, 1981.
[51]H. Rafii-Tabar, "Computational modelling of thermo-mechanical and transport properties of carbon nanotubes," Physics Reports, vol. 390, no. 4-5, pp. 235-452, 2004.
[52]A. Gannepalli and S. K. Mallapragada, "Molecular dynamics studies of plastic deformation during silicon nanoindentation," Nanotechnology, vol. 12, no. 3, p. 250, 2001.
[53]F. Shimizu, S. Ogata, and J. Li, "Theory of shear banding in metallic glasses and molecular dynamics calculations," Materials transactions, vol. 48, no. 11, pp. 2923-2927, 2007.
[54]C. Chen, Y. Shi, Y. S. Zhang, J. Zhu, and Y. Yan, "Size dependence of Young’s modulus in ZnO nanowires," Physical review letters, vol. 96, no. 7, p. 075505, 2006.
[55]D. Frenkel and J.-P. Hansen, "Understanding liquids: A computer game?," Physics world, vol. 9, no. 4, p. 35, 1996.
[56]M. P. Allen and D. J. Tildesley, Computer simulation of liquids. Oxford university press, 2017.
[57]D. C. Rapaport and D. C. R. Rapaport, The art of molecular dynamics simulation. Cambridge university press, 2004.
[58]J. M. Haile, "Molecular dynamics simulation," Elementary methods, 1992.
[59]P. Sharma et al., "Ferromagnetism above room temperature in bulk and transparent thin films of Mn-doped ZnO," Nature materials, vol. 2, no. 10, p. 673, 2003.
[60]H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and S.-C. Zhang, "Topological insulators in Bi 2 Se 3, Bi 2 Te 3 and Sb 2 Te 3 with a single Dirac cone on the surface," Nature physics, vol. 5, no. 6, p. 438, 2009.
[61]C. Kittel, Introduction to solid state physics. Wiley New York, 1976.
[62]B.-J. Lee, J.-H. Shim, and M. J. P. R. B. Baskes, "Semiempirical atomic potentials for the fcc metals Cu, Ag, Au, Ni, Pd, Pt, Al, and Pb based on first and second nearest-neighbor modified embedded atom method," vol. 68, no. 14, p. 144112, 2003.
[63]B.-J. Lee, M. Baskes, H. Kim, and Y. K. J. P. R. B. Cho, "Second nearest-neighbor modified embedded atom method potentials for bcc transition metals," vol. 64, no. 18, p. 184102, 2001.
[64]Y.-M. Kim, B.-J. Lee, and M. J. P. R. B. Baskes, "Modified embedded-atom method interatomic potentials for Ti and Zr," vol. 74, no. 1, p. 014101, 2006.
[65]J.-S. Kim, D. Seol, J. Ji, H.-S. Jang, Y. Kim, and B.-J. J. C. Lee, "Second nearest-neighbor modified embedded-atom method interatomic potentials for the Pt-M (M= Al, Co, Cu, Mo, Ni, Ti, V) binary systems," vol. 59, pp. 131-141, 2017.
[66]S. J. A. A. Wang, "Paracrystalline property of high-entropy alloys," vol. 3, no. 10, p. 102105, 2013.
[67]O. Senkov, J. Jensen, A. Pilchak, D. Miracle, H. J. M. Fraser, and Design, "Compositional variation effects on the microstructure and properties of a refractory high-entropy superalloy AlMo0. 5NbTa0. 5TiZr," vol. 139, pp. 498-511, 2018.
[68]D. J. Evans and B. L. Holian, "The nose–hoover thermostat," The Journal of chemical physics, vol. 83, no. 8, pp. 4069-4074, 1985.
[69]W. C. Swope, H. C. Andersen, P. H. Berens, and K. R. Wilson, "A computer simulation method for the calculation of equilibrium constants for the formation of physical clusters of molecules: Application to small water clusters," The Journal of Chemical Physics, vol. 76, no. 1, pp. 637-649, 1982.