本文利用Force-Matching (FM)方法為基礎，並結合Basin-Hopping (BH)方法，發展一套可擬合出各種多元素金屬材料勢能參數的方法。其間以密度泛函理論所計算的數值當目標值，而FM和BH兩種方法的結合不僅可以準確及快速的擬合所需之勢能參數，更將擬合後的勢能應用於分子力學模擬，以求得合理之結構及其機械性質。此方法擬合出的勢能參數，將能克服以往金屬玻璃材料在分子模擬與實驗上結果誤差較大的缺點。
Tight-Binding函數為所選擇的勢能函數能有效的描述過渡金屬元素間的交互作用關係；在驗證參數方面，選擇MgZnCa金屬玻璃為研究模擬材料，由於原子間排列組合交互作用複雜，故相對應之勢能參數擬合更為困難，包含Mg-Mg、Zn-Zn、Ca-Ca、Mg-Zn、Mg-Ca及Zn-Ca六種不同勢能參數組。利用FM結合BH的方法將可精確有效地擬合出此六種不同原子間的最佳勢能參數。最後以分子動力學應用此勢能參數，探討MgZnCa 金屬玻璃材料的機械性質，並與實驗結果作分析及比較。

This research will combine the Force-Matching method and Basin-Hopping method to develop a new method for fitting the potential function parameters of the multi-element alloys. The combination of Force-Matching and Basin-Hopping methods not only can give the fitting parameters accurately and rapidly, but also can be used in the molecular mechanics to obtain a reasonable structure and mechanical properties of material. The new parameters obtained by this fitting approach are able to decrease the relative errors between simulation and experiment of metallic glass.
The choice of potential function is Tight-binding function that can describe the interaction of transition metallic atoms accurately. The MgZnCa bulk metallic glass is chose to verify the parameters for simulation. Owing to the high complexity of the interaction between atoms it is more difficult to fit the potential parameters for the respective pair, including the potential parameters of Mg-Mg, Zn-Zn, Ca-Ca, Mg-Zn, Mg-Ca and Zn-Ca interactions. The combination of FM and BH methods can fit out the six potential parameters accurately and rapidly. Those parameters will be used in molecular dynamics to investigate the mechanical properties and compared with experiment values.

目錄 i
圖次 iv
表次 vi
中文摘要 vii
Abstract viii
第一章 序論 1
1.1研究目的與動機 1
1.2文獻回顧 5
1.3論文架構 8
第二章 理論基礎及方法 9
2.1密度泛函理論(DENSITY FUNCTIONAL THEORY) 9
2.1.1 Thomas-Fermi 理論 9
2.1.2 Hohenberg-Kohn 理論 10
2.1.3 Kohn-Sham方程式 10
2.1.4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擬合勢能參數 14
2.2.3 Big-Bang (BB) method 16
2.2.4 Basin-Hopping (BH) method 17
2.3分子靜力學理論 19
2.3.1 LBFGS法 20
2.3.2火炎演算法 22
2.3.3共軛梯度法 23
2.4分子動力學理論 24
2.4.1積分法則 25
2.4.2諾斯-胡佛恆溫法(Nosé-Hoover thermostat) 26
2.4.3時間步階選取 28
2.5原子級應力分析 29
2.6結構分析 32
2.6.1 HA鍵型指數法 32
2.6.2區域應變分析 33
2.7週期邊界的處理 34
2.8鄰近原子表列數值方法 35
2.8.1截斷半徑法 36
2.8.2 Verlet List表列法 37
2.8.3 Cell Link表列法 39
2.8.4 Verlet List結合Cell Link表列法 41
2.9 模擬流程 42
第三章 結果與討論 45
3.1 擬合勢能參數 45
3.1.1 尋找最佳DFT設定及建立reference data 45
3.1.2 參數擬合結果 47
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 56
3.3 拉伸模擬試驗 63
3.3.1 拉伸模型建立 63
3.3.2試驗結果分析 64
第四章 結論與建議 73
4.1 結論 73
4.2 建議與未來展望 75
參考文獻 76

　[1] N. M. Pugno, “On the strength of the carbon nanotube-based space elevator cable: from nanomechanics to megamechanics,” Journal of Physics-Condensed Matter, vol. 18, pp. S1971-S1990, 2006.
　[2] N. R. Jana, Y. F. Chen, and X. G. Peng, “Size- and shape-controlled magnetic (Cr, Mn, Fe, Co, Ni) oxide nanocrystals via a simple and general approach,” Chemistry of Materials, vol. 16, pp. 3931-3935, 2004.
　[3] W. Z. Li, W. J. Zhou, H. Q. Li, Z. H. Zhou, B. Zhou, G. Q. Sun, and Q. Xin, “Nano-stuctured Pt-Fe/C as cathode catalyst in direct methanol fuel cell,” Electrochimica Acta, vol. 49, pp. 1045-1055, 2004.
　[4] G. Shao, “Prediction of structural stabilities of transition-metal disilicide alloys by the density functional theory,” Acta Materialia, vol. 53, pp. 3729-3736, 2005.
　[5] W. He, A. Chuang, Z. Cao, and P. K. Liaw, “Biocompatibility Study of Zirconium-Based Bulk Metallic Glasses for Orthopedic Applications,” Metallurgical and Materials Transactions a-Physical Metallurgy and Materials Science, vol. 41A, pp. 1726-1734, 2010.
　[6] B. Zberg, P. J. Uggowitzer, and J. F. Loffler, “MgZnCa glasses without clinically observable hydrogen evolution for biodegradable implants,” Nature Materials, vol. 8, pp. 887-891, 2009.
　[7] Y. B. Wang, X. H. Xie, H. F. Li, X. L. Wang, M. Z. Zhao, E. W. Zhang, Y. J. Bai, Y. F. Zheng, and L. Qin, “Biodegradable CaMgZn bulk metallic glass for potential skeletal application,” Acta Biomaterialia, vol. 7, pp. 3196-3208, 2011.
　[8] W. Klement, R. H. Willens, and P. Duwez, “Non-Crystalline Structure in Solidified Gold-Silicon Alloys,” Nature, vol. 187, pp. 869-870, 1960.
　[9] A. L. Greer, “Metallic Glasses,” Science, vol. 267, pp. 1947-1953, 1995.
　[10] C. C. Hays, C. P. Kim, and W. L. Johnson, “Microstructure controlled shear band pattern formation and enhanced plasticity of bulk metallic glasses containing in situ formed ductile phase dendrite dispersions,” Physical Review Letters, vol. 84, pp. 2901-2904, 2000.
　[11] J. Schroers and W. L. Johnson, “Ductile bulk metallic glass,” Physical Review Letters, vol. 93, 2004.
　[12] S. Xie and E. P. George, “Hardness and shear band evolution in bulk metallic glasses after plastic deformation and annealing,” Acta Materialia, vol. 56, pp. 5202-5213, 2008.
　[13] D. Sopu, Y. Ritter, H. Gleiter, and K. Albe, “Deformation behavior of bulk and nanostructured metallic glasses studied via molecular dynamics simulations,” Physical Review B, vol. 83, 2011.
　[14] Y. Shen and G. P. Zheng, “Modeling of shear band multiplication and interaction in metallic glass matrix composites,” Scripta Materialia, vol. 63, pp. 181-184, 2010.
　[15] Q. K. Li and M. Li, “Molecular dynamics simulation of intrinsic and extrinsic mechanical properties of amorphous metals,” Intermetallics, vol. 14, pp. 1005-1010, 2006.
　[16] Y. G. Xu and G. R. Liu, “Fitting interatomic potentials using molecular dynamics simulations and inter-generation projection genetic algorithm,” Journal of Micromechanics and Microengineering, vol. 13, pp. 254-260, 2003.
　[17] N. I. Papanicolaou, G. C. Kallinteris, G. A. Evangelakis, D. A. Papaconstantopoulos, and M. J. Mehl, “Second-moment interatomic potential for Cu-Au alloys based on total-energy calculations and its application to molecular-dynamics simulations,” Journal of Physics-Condensed Matter, vol. 10, pp. 10979-10990, 1998.
　[18] C. Barreteau, D. Spanjaard, and M. C. Desjonqueres, “Electronic structure and total energy of transition metals from an spd tight-binding method: Application to surfaces and clusters of Rh,” Physical Review B, vol. 58, pp. 9721-9731, 1998.
　[19] C. M. Goringe, D. R. Bowler, and E. Hernandez, “Tight-binding modelling of materials,” Reports on Progress in Physics, vol. 60, pp. 1447-1512, 1997.
　[20] R. E. Cohen, M. J. Mehl, and D. A. Papaconstantopoulos, “Tight-Binding Total-Energy Method for Transition and Noble-Metals,” Physical Review B, vol. 50, pp. 14694-14697, 1994.
　[21] M. M. Sigalas and D. A. Papaconstantopoulos, “Transferable Total-Energy Parametrization for Metals - Applications to Elastic-Constant Determination,” Physical Review B, vol. 49, pp. 1574-1579, 1994.
　[22] F. Ercolessi and J. B. Adams, “Interatomic Potentials from 1st-Principles Calculations - the Force-Matching Method,” Europhysics Letters, vol. 26, pp. 583-588, 1994.
　[23] O. Akin-Ojo, Y. Song, and F. Wang, “Developing ab initio quality force fields from condensed phase quantum-mechanics/molecular-mechanics calculations through the adaptive force matching method,” Journal of Chemical Physics, vol. 129, 2008.
　[24] Y. H. Li, D. J. Siegel, J. B. Adams, and X. Y. Liu, “Embedded-atom-method tantalum potential developed by the force-matching method,” Physical Review B, vol. 67, 2003.
　[25] G. Grochola, S. P. Russo, and I. K. Snook, “On fitting a gold embedded atom method potential using the force matching method,” Journal of Chemical Physics, vol. 123, 2005.
　[26] X. Y. Liu, P. P. Ohotnicky, J. B. Adams, C. L. Rohrer, and R. W. Hyland, “Anisotropic surface segregation in Al-Mg alloys,” Surface Science, vol. 373, pp. 357-370, 1997.
　[27] A. Landa, P. Wynblatt, D. J. Siegel, J. B. Adams, O. N. Mryasov, and X. Y. Liu, “Development of glue-type potentials for the Ai-Pb system: Phase diagram calculation,” Acta Materialia, vol. 48, pp. 1753-1761, 2000.
　[28] X. L. Ding, Z. Y. Li, J. L. Yang, J. G. Hou, and Q. S. Zhu, “Adsorption energies of molecular oxygen on Au clusters,” Journal of Chemical Physics, vol. 120, pp. 9594-9600, 2004.
　[29] X. R. Zhang, X. L. Ding, B. Dai, and J. L. Yang, “Density functional theory study of W-n (n=2-4) clusters,” Journal of Molecular Structure-Theochem, vol. 757, pp. 113-118, 2005.
　[30] D. A. Stewart, “Magnetism in coaxial palladium nanowires,” Journal of Applied Physics, vol. 101, 2007.
　[31] Z. G. Zhu, A. Chutia, R. Sahnoun, M. Koyama, H. Tsuboi, N. Hatakeyama, A. Endou, H. Takaba, M. Kubo, C. A. Del Carpio, and A. Miyamoto, “Theoretical study on electronic and electrical properties of nanostructural ZnO,” Japanese Journal of Applied Physics, vol. 47, pp. 2999-3006, 2008.
　[32] B. Arman, C. Brandl, S. N. Luo, T. C. Germann, A. Misra, and T. Cagin, “Plasticity in Cu(111)/Cu46Zr54 glass nanolaminates under uniaxial compression,” Journal of Applied Physics, vol. 110, 2011.
　[33] X. D. Wang, H. B. Lou, J. Bednarcik, H. Franz, H. W. Sheng, Q. P. Cao, and J. Z. Jiang, “Structural evolution in bulk metallic glass under high-temperature tension,” Applied Physics Letters, vol. 102, 2013.
　[34] A. Franceschetti, S. J. Pennycook, and S. T. Pantelides, “Oxygen chemisorption on Au nanoparticles,” Chemical Physics Letters, vol. 374, pp. 471-475, 2003.
　[35] G. H. Ryu, S. C. Park, and S. B. Lee, “Molecular orbital study of the interactions of CO molecules adsorbed on a W(111) surface,” Surface Science, vol. 427-28, pp. 419-425, 1999.
　[36] P. Hohenberg and W. Kohn, “INHOMOGENEOUS ELECTRON GAS,” Physical Review B, vol. 136, pp. B864-&, 1964.
　[37] W. Kohn and L. J. Sham, “Self-Consistent Equations Including Exchange and Correlation Effects,” Physical Review, vol. 140, pp. 1133-&, 1965.
　[38] V. Rosato, M. Guillope, and B. Legrand, “Thermodynamical and Structural-Properties of Fcc Transition-Metals Using a Simple Tight-Binding Model,” Philosophical Magazine a-Physics of Condensed Matter Structure Defects and Mechanical Properties, vol. 59, pp. 321-336, 1989.
　[39] M. S. Daw and M. I. Baskes, “Embedded-Atom Method - Derivation and Application to Impurities, Surfaces, and Other Defects in Metals,” Physical Review B, vol. 29, pp. 6443-6453, 1984.
　[40] S. M. Foiles, M. I. Baskes, and M. S. Daw, “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, 1986.
　[41] D. J. Wales and J. P. K. Doye, “Global optimization by basin-hopping and the lowest energy structures of Lennard-Jones clusters containing up to 110 atoms,” Journal of Physical Chemistry A, vol. 101, pp. 5111-5116, 1997.
　[42] K. A. Jackson, M. Horoi, I. Chaudhuri, T. Frauenheim, and A. A. Shvartsburg, “Unraveling the shape transformation in silicon clusters,” Physical Review Letters, vol. 93, 2004.
　[43] D. C. Liu and J. Nocedal, “On the Limited Memory Bfgs Method for Large-Scale Optimization,” Mathematical Programming, vol. 45, pp. 503-528, 1989.
　[44] S. Hamad, C. R. A. Catlow, S. M. Woodley, S. Lago, and J. A. Mejias, “Structure and stability of small TiO2 nanoparticles,” Journal of Physical Chemistry B, vol. 109, pp. 15741-15748, 2005.
　[45] W. W. Hager and H. C. Zhang, “A new conjugate gradient method with guaranteed descent and an efficient line search,” Siam Journal on Optimization, vol. 16, pp. 170-192, 2005.
　[46] L. Zhang, W. J. Zhou, and D. H. Li, “Some descent three-term conjugate gradient methods and their global convergence,” Optimization Methods & Software, vol. 22, pp. 697-711, 2007.
　[47] W. Quapp, “A growing string method for the reaction pathway defined by a Newton trajectory,” Journal of Chemical Physics, vol. 122, 2005.
　[48] C. G. Broyden, “Convergence of Single-Rank Quasi-Newton Methods,” Mathematics of Computation, vol. 24, pp. 365-&, 1970.
　[49] R. Fletcher, “A New Approach to Variable Metric Algorithms,” Computer Journal, vol. 13, pp. 317-&, 1970.
　[50] D. Goldfarb, “A Family of Variable-Metric Methods Derived by Variational Means,” Mathematics of Computation, vol. 24, pp. 23-&, 1970.
　[51] D. F. Shanno, “Conditioning of Quasi-Newton Methods for Function Minimization,” Mathematics of Computation, vol. 24, pp. 647-&, 1970.
　[52] R. H. Byrd, P. H. Lu, J. Nocedal, and C. Y. Zhu, “A Limited Memory Algorithm for Bound Constrained Optimization,” Siam Journal on Scientific Computing, vol. 16, pp. 1190-1208, 1995.
　[53] P. A. T. Olsson, S. Melin, and C. Persson, “Atomistic simulations of tensile and bending properties of single-crystal bcc iron nanobeams,” Physical Review B, vol. 76, 2007.
　[54] E. Bitzek, P. Koskinen, F. Gahler, M. Moseler, and P. Gumbsch, “Structural relaxation made simple,” Physical Review Letters, vol. 97, 2006.
　[55] N. Andrei, “Scaled memoryless BFGS preconditioned conjugate gradient algorithm for unconstrained optimization,” Optimization Methods & Software, vol. 22, pp. 561-571, 2007.
　[56] B. J. O. Teleman, S. Engstrom, “A molecular-dynamics simulation of a water model with intramolecular defgrees of freedom,” Mol. Phys., vol. 60, pp. 193-203, 1987.
　[57] S. Nose, “A unified formulation of the constant temperature molecular-dynamics methods,” J. Chem. Phys., vol. 81, pp. 511-519, 1984.
　[58] S. C. M.S. Lee, D.G. Kanhere, “First-principles investigation of finite-temperature behavior in small sodium clusters,” J. Chem. Phys., vol. 123, 2005.
　[59] M. S. L. S.M. Ghazi, D.G. 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,” J. Chem. Phys., vol. 128, 2008.
　[60] S. N. a. C. S. N. Chandra, “Local elastic properties of carbon nanotubes in the presence of tone-wales defects,” Physical Review B, vol. 69, p. 094101, 2004.
　[61] M. H. N. Tokita, C. Azuma and T. Dotera, “Voronoi space division of a polymer: Topological effects, free volume, and surface end segregation,” Journal of Chemical Physics, vol. 120, pp. 496-505, 2004.
　[62] 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, 2004.
　[63] N. Tokita, M. Hirabayashi, C. Azuma, and T. Dotera, “Voronoi space division of a polymer: Topological effects, free volume, and surface end segregation,” Journal of Chemical Physics, vol. 120, pp. 496-505, 2004.
　[64] D. Srolovitz, K. Maeda, V. Vitek, and T. Egami, “Structural Defects in Amorphous Solids Statistical-Analysis of a Computer-Model,” Philosophical Magazine a-Physics of Condensed Matter Structure Defects and Mechanical Properties, vol. 44, pp. 847-866, 1981.
　[65] N. Miyazaki and Y. Shiozaki, “Calculation of mechanical properties of solids using molecular dynamics method,” Jsme International Journal Series a-Mechanics and Material Engineering, vol. 39, pp. 606-612, 1996.
　[66] H. Rafii-Tabar, “Computational modelling of thermo-mechanical and transport properties of carbon nanotubes (vol 390, pg 235, 2004),” Physics Reports-Review Section of Physics Letters, vol. 394, pp. 357-357, 2004.
　[67] J. D. Honeycutt and H. C. Andersen, “Molecular-Dynamics Study of Melting and Freezing of Small Lennard-Jones Clusters,” Journal of Physical Chemistry, vol. 91, pp. 4950-4963, 1987.
　[68] A. Gannepalli and S. K. Mallapragada, “Molecular dynamics studies of plastic deformation during silicon nanoindentation,” Nanotechnology, vol. 12, pp. 250-257, 2001.
　[69] A. J. Cao, Y. Q. Cheng, and E. Ma, “Structural processes that initiate shear localization in metallic glass,” Acta Materialia, vol. 57, pp. 5146-5155, 2009.
　[70] F. Shimizu, S. Ogata, and J. Li, “Theory of shear banding in metallic glasses and molecular dynamics calculations,” Materials Transactions, vol. 48, pp. 2923-2927, 2007.
　[71] C. Q. Chen, Y. Shi, Y. S. Zhang, J. Zhu, and Y. J. Yan, “Size dependence of Young's modulus in ZnO nanowires,” Physical Review Letters, vol. 96, 2006.
　[72] D. Frenkel and J. P. Hansen, “Understanding liquids: A computer game?,” Physics World, vol. 9, pp. 35-40, 1996.
　[73] M. P. Allen and D. J. Tildesley, Computer simulation of liquids: Oxford university press, 1989.
　[74] D. C. Rapaport, The art of molecular dynamics simulation: Cambridge university press, 2004.
　[75] J. M. Haile, Molecular dynamics simulation: elementary methods: John Wiley & Sons, Inc., 1992.
　[76] P. Srepusharawoot, C. M. Araujo, A. Blomqvist, R. H. Scheicher, and R. Ahuja, “A comparative investigation of H(2) adsorption strength in Cd- and Zn-based metal organic framework-5,” Journal of Chemical Physics, vol. 129, 2008.
　[77] H. Y. Yan, Y. Q. Li, Y. R. Guo, Q. G. Song, and Y. F. Chen, “Ferromagnetic properties of Cu-doped ZnS: A density functional theory study,” Physica B-Condensed Matter, vol. 406, pp. 545-547, 2011.
　[78] L. Lin, P. Liang, L. Yang, L. J. Chen, Z. Liu, and Y. M. Wang, “Phase stability comparison by first principle calculation and experimental observation of microstructure evolution in a Mg-6Gd-2Zn(wt%) alloy,” Materials Science and Engineering a-Structural Materials Properties Microstructure and Processing, vol. 527, pp. 2643-2648, 2010.
　[79] Y. Zhong, K. Ozturk, J. O. Sofo, and Z. K. Liu, “Contribution of first-principles energetics to the Ca-Mg thermodynamic modeling,” Journal of Alloys and Compounds, vol. 420, pp. 98-106, 2006.
　[80] Y. H. Sun, Q. Zhao, J. Y. Gao, Y. Ye, W. Wang, R. Zhu, J. Xu, L. Chen, J. Yang, L. Dai, Z. M. Liao, and D. P. Yu, “In situ growth, structure characterization, and enhanced photocatalysis of high-quality, single-crystalline ZnTe/ZnO branched nanoheterostructures,” Nanoscale, vol. 3, pp. 4418-4426, 2011.
　[81] R. Arroyave, A. van de Walle, and Z. K. Liu, “First-principles calculations of the Zn-Zr system,” Acta Materialia, vol. 54, pp. 473-482, 2006.
　[82] J. P. Perdew and Y. Wang, “Accurate and Simple Analytic Representation of the Electron-Gas Correlation-Energy,” Physical Review B, vol. 45, pp. 13244-13249, 1992.
　[83] J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized gradient approximation made simple,” Physical Review Letters, vol. 77, pp. 3865-3868, 1996.
　[84] F. Cleri and V. Rosato, “Tight-Binding Potentials for Transition-Metals and Alloys,” Physical Review B, vol. 48, pp. 22-33, 1993.
　[85] M. A. Karolewski, “Tight-binding potentials for sputtering simulations with fcc and bcc metals,” Radiation Effects and Defects in Solids, vol. 153, pp. 239-255, 2001.
　[86] C. Kittel and P. McEuen, Introduction to solid state physics vol. 7: Wiley New York, 1996.
　[87] G. Simmons, “Single crystal elastic constants and calculated aggregate properties,” DTIC Document1965.
　[88] P. Pyykko and M. Atsumi, “Molecular Double-Bond Covalent Radii for Elements Li-E112,” Chemistry-a European Journal, vol. 15, pp. 12770-12779, 2009.
　[89] Y. Qi, T. Cagin, Y. Kimura, and W. A. Goddard, “Molecular-dynamics simulations of glass formation and crystallization in binary liquid metals: Cu-Ag and Cu-Ni,” Physical Review B, vol. 59, pp. 3527-3533, 1999.
　[90] B. Zberg, E. R. Arata, P. J. Uggowitzer, and J. F. Loffler, “Tensile properties of glassy MgZnCa wires and reliability analysis using Weibull statistics,” Acta Materialia, vol. 57, pp. 3223-3231, 2009.
　[91] F. Qin, G. Xie, Z. Dan, S. Zhu, and I. Seki, “Corrosion behavior and mechanical properties of Mg–Zn–Ca amorphous alloys,” Intermetallics, vol. 42, pp. 9-13, 2013.
　[92] M. W. Chen, “Mechanical behavior of metallic glasses: Microscopic understanding of strength and ductility,” Annual Review of Materials Research, vol. 38, pp. 445-469, 2008.