CGMD ( Coarse Grained Molecular Dynamics )是一種能應用在多重尺度模擬上的方法，該法其簡化節點的方式主要利用有限元素法(FEM)權重函數(Weighting function)的概念，而運動方程式的推導是由分子動力學(Molecular Dynamics)經統計力學處理所求得，因此在探討多重尺度模擬的問題時，模擬系統中不同尺度仍處於相同理論基礎。不同尺度下之勢能函數可應用修正勢能函數方法，而跨尺度模擬時介面所產生的不連續問題，其問題在本研究中也有解決應對方法。

本研究中除將CGMD理論完整推導外，更將CGMD應用模擬微觀銅結構(001)面之奈米壓痕行為及其現象，用以量測不同材料的機械性質以及探討各種溫度、基板等效應，並將其得到結果與分子動力學比較。

經過模擬分析後，發現CGMD模擬結果與分子動力學非常相似，可從四種方向觀察得到：(a)壓痕力量與深度關係圖(b)奈米壓痕所求得之硬度與楊氏模數值(c)奈米壓痕物理現象(d)奈米壓痕缺陷結構。在本研究中的CGMD電腦數值模擬時間大約為分子動力學的1/3~1/4。

Coarse Grained Molecular Dynamics (CGMD) is a technique of multiscale simulations. The concepts of CGMD utilize the weighting function of Finite Element Method (FEM) for reduced nodes. And the equations of motion are derived from Molecular Dynamics through statistical mechanics. Therefore, the simulated systems based on the same theories for discussing multiscale simulations’ problems. Applications of modified potential function on different scales. And this research solves the problems for seamless coupling of length scales.

This paper not only derives the theories of CGMD completely, but alos applies on Cu(001) nanoindentation. And it makes use of measuring the mechanical properties of different materials and discussing the effects of temperature, substrate etc.. Moreover, the results of CGMD compared with Molecular Dynamics.

After the analysis of simulations, the results of CGMD and Molecular Dynamics are well consistent illustrated as below : (a) force-displacement curve. (b) hardness and Young’s modulus. (c) the physical behaviors of nanoindentation. (d) defect structures of nanoindentation. Time-consuming for CGMD costs 1/3~1/4 by comparing with Molecular Dynamics.

[1] R. E. Rudd and J. Q. Broughton, “Coarse-grained Molecular Dynamics and Atomic Limit of Finite Elements”, Phys. Rev. B, Vol. 58, No. 10, pp. R5893-R5896, 1998.
[2] E. B. Tadmor and Rob Phillips, “Mixed Atomistic and Continuum Models of Deformation in Solids”, Langmuir, 12, pp. 4529-4534, 1996.
[3] Jeremy Q. Broughton, Farid F. Abraham, Noam Bernstein and Efthimios Kaxiras, “Concurrent coupling of length scales: Methodology and application”, Phys. Rev. B, Vol. 60, No. 4, pp. 2391-2403, 1999.
[4] J. Knap and M. Ortiz, “An analysis of the quasicontinuum method”, J. Mech. Phys. Solids, 49, pp. 1899-1923, 2001.
[5] Elefterios Lidorikis, Martina E. Bachlechner, Rajiv K. Kalia, Aiichiro Nakano, Priya Vashishta and George Z. Voyiadjis, “Coupling Length Scales for Multiscale Atomistics-Continuum Simulations: Atomistically Induced Stress Distributions in Si/Si3N4 Nanopixels”, Phys. Rev. Lett., Vol. 87, No. 8, pp. 086104-1-086104-4, 2001.
[6] S. Curtarolo and G. Ceder, “Dynamics of an Inhomogeneously Coarse Grained Multiscale System”, Phys. Rev. Lett., Vol. 88, No. 25, pp. 255504-1-255504-4, 2002.
[7] D. J. Diestler, “Coarse-grained descriptions of multiple scale processes in solid systems”, Phys. Rev. B, 66, pp. 184104-1-184104-7, 2002.
[8] Gregory J. Wagner and Wing Kam Liu, “Coupling of atomistic and continuum simulations using a bridging scale decomposition”, J. Comput. Phys., 190, pp. 249-274, 2003.
[9] Xiantao Li and Weinan E, “Multiscale modeling of the dynamics of solids at finite temperature”, J. Mech. Phys. Solids, 53, pp. 1650-1685, 2005.
[10] L. M. Dupuy, E. B. Tadmor, R. E. Miller and R. Phillips, “Finite Temperature Quasicontinuum: Molecular Dynamics without All the Atoms”, Phys. Rev. Lett., Vol. 95, No. 6, pp. 060202-1-060202-4, 2005.
[11] P. A. Klein and J. A. Zimmerman, “Coupled atomistic-continuum simulations using arbitrary overlapping domains”, J. Comput. Phys., 213, pp. 86-116, 2006.
[12] Edward J. Smiley, Zbigniew Postawa, Igor A. Wojciechowski, Nicholas Winograd and Barbara J. Garrison, “Coarse-grained molecular dynamics studies of cluster-bombarded benzene crystals”, Appl. Surf. Sci., 252, pp. 6436-6439, 2006.
[13] U. Landman, W.D. Luedtke, N.A. Burnham and R.J. Colton, “Atomistic Mechanisms and Dynamics of Adhesion, Nanoindentation and Fracture”, Science, Vol. 248, No. 4954, pp. 454-461, 1990.
[14] W. C. Oliver and G. M. Pharr, “An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments”, J. Mater. Res., 7, pp. 1564-1583, 1992.
[15] J. Shimizu, H. Eda, M. Yoritsune and E. Ohmura, “Molecular dynamics simulation of friction on the atomic scale”, Nanotechnology, 9, No. 2, pp. 118-123, 1998.
[16] Cynthia L. Kelchner, S. J. Plimpton and J. C. Hamilton, “Dislocation nucleation and defect structure during surface indentation”, Phys. Rev. B, Vol. 58, No. 17, pp. 11 085-11 088, 1998.
[17] F. N. Dzegilenko, D. Srivastava and S. Saini, “Nanoscale etching and indentation of a silicon(001) surface with carbon nanotube tips”, Nanotechnology, 11, N0.3, pp. 253-257, 1999.
[18] T. H. Fang, “Study on the Nano-scale Processing Technology Using Atomic Force Microscopy”, Ph.D. Dissertation, National Cheng Kung University, Tainan, Taiwan, 2000.
[19] G. S. Smith, E. B. Tadmor, N. Bernstein and E. Kaxiras, “Multiscale Simulations of Silicon Nanoindentation”, Acta Mater., 49, pp. 4089-4101, 2001.
[20] J. Knap and M. Ortiz, “Effect of Indenter-Radius Size on Au(001) Nanoindentation”, Phys. Rev. Lett., Vol. 90, No. 22, pp. 226102-1- 226102-4, 2003.
[21] J. Chen, W. Wang, L. H. Qian and K. Lu, “Critical shear stress for onset of plasticity in a nanocrystalline Cu determined by using nanoindentation”, Scripta Mater., 49, pp. 645-650, 2003.
[22] Jianghong Gong, Hezhuo Miao and Zhijian Peng, “Analysis of the nanoindentation data measured with a Berkovich indenter for brittle materials: effect of the residual contact stress”, Acta Mater., 52, pp. 785- 793, 2003.
[23] Denis Saraev and Ronald E Miller, “Atomistic simulation of nanoindentation into copper multilayers”, Model. Simul. Mater. Sci. Eng., 13, pp. 1089-1099, 2005.
[24] Kisaragi Yashiro, Atsushi Furuta and Yoshihiro Tomita, “Nanoindentation on crystal/amorphous polyethylene: Molecular dynamics study”, Comput. Mater. Sci., 38, pp. 136-143, 2006.
[25] D E Kim and S I Oh, “Atomistic simulation of structural phase transformations in monocrystalline silicon induced by nanoindentation”, Nanotechnology, 17, pp. 2259-2265, 2006.
[26] 吳書帆, “多重尺度CGMD在奈米壓痕的應用”, 國立成功大學, 碩士論文, 2006.
[27] J. E. Lennard-Jones, Proc. Roy. Soc. Lond., 1924.
[28] L. A. Girifalco and V. G. Weizer, “Application of the Morse Potential Function to Cubic Metals”, Phys. Rev., Vol. 114, No. 3, pp. 687-690, 1959.
[29] 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”, Phys. Rev. B, Vol. 33, No. 12, pp. 7983-7911, 1986.
[30] M. I. Baskes, J. S. Nelson and A. F. Wright, “Semiempirical modified embedded-atom potentials for silicon and germanium”, Phys. Rev. B, Vol. 40, No. 9, pp. 6085-6100, 1989.
[31] M. I. Baskes, “Modified embedded-atom potentials for cubic materials and impurities”, Phys. Rev. B, Vol. 46, No. 5, pp. 2727-2742, 1992.
[32] M. S. Daw, S. M. Faoiles and M. I. Baskes, “The embedded-atom method: a review of theory and applications”, Material Science Report., Vol. 9, pp. 251-310, 1993.
[33] 陳道隆, “量子系統之混合表象與高度應變率對奈米金線力學行為之影響”, 國立成功大學, 博士論文, 2005.
[34] J. M. Hailie, “Molecular Dynamics Simulation Elementary Methods”, New York: John Wiley & Sons. Inc., 1993.
[35] T. Iwaki, “Molecular Dynamic Study on Stress-Strain in Very Thin Film”, JSME Int. J. Ser. A, Vol. 39, No. 3, pp. 346-353, 1988.
[36] 陳冠維, “掃描式探針顯微鏡摩擦力檢測應用於不鏽鋼氮離子植佈技術之建立及材料微奈米機械性質檢測”, 國立成功大學, 碩士論文, 2003.
[37] I. N. Sneddon, “The relation between load and penetration in the axisymmetric boussinesq problem for a punch of arbitrary profile”, Int. J. Eng. Sci., Vol. 3, pp. 47-53, 1965.
[38] R. B. King, “Elastic analysis of some punch problems for a layered medium”, Int. J. Solids Struct., Vol. 23, pp. 1657-1664, 1987.
[39] J. B. Pethica, R. Hutchings and W. C. Oliver, Phil. Mag., Vol. 48, pp. 593 -597, 1983.
[40] Ju Li, Krystyn J. Van Vliet, Ting Zhu, Sidney Yip and Subra Suresh, “Atomistic mechanisms governing elastic limit and incipient plasticity in crystals”, Nature, Vol. 418, pp. 307-310, 2002.
[41] A. Dawid, Z. Dendzik and Z. Gburski, “Molecular dynamics study of ultrathin argon layer covering fullerene molecule”, J. Mol. Struct., 704, pp. 173-176, 2004.
[42] Te-Hua Fang, Cheng-I Weng and Jee-Gong Chang, “Molecular dynamics analysis of temperature effects on nanoindentation measurement”, Mater. Sci. Eng., A357, pp. 7-12, 2003.
[43] 林彥宏, “奈米壓痕之表面層效應與結構變化探討”, 國立中正大學, 碩士論文, 2005.
[44] D. Hull and D. J. Bacon, “Introduction to Dislocations”, University of Liverpool, UK, 1984.
[45] K. Ohno, K. Esfarjani, Y. Kawazoe, “Computational Materials Science From Ab Initio to Monte Carlo Methods”, Springer, 1999.
[46] 饒智昇, “以分子動力學方法研究奈米級微結構的力學問題”, 國立成功大學, 碩士論文, 1999.
[47] 陳道隆, “以分子動力學研究奈米級微結構之拉伸、壓縮、扭轉變形機制”, 國立成功大學, 碩士論文, 2001.
[48] 廖英博, “耦合原子尺度模擬與連體描述之三維擬連體法理論與實作”, 國立臺灣大學, 碩士論文, 2004.