石墨烯薄膜杨氏模量的分子动力学研究
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摘要
石墨烯具有出色的光学、电学、磁学及力学方面的性质,有着广阔的应用前景,可被用于质量和气体传感器、微型机械发电机、纳米复合材料、晶体管、半导体设备、纳米微电子机械系统等领域。而石墨烯的力学性质将直接影响到其各方面性能的发挥,也制约着石墨烯纳米装置的稳定性和可靠性。
本文采用分子动力学(MD)方法利用圆膜弹性理论对独立式悬置圆膜石墨烯进行模拟计算获得石墨烯的杨氏模量。研究的主要内容如下:
1.按扰度的大小的不同采用分阶段研究的方法研究了单层石墨烯的杨氏模量。薄膜在扰度较小时和扰度较大时具有不同的弹性反应特征。在扰度较小的情况下,压头对薄膜形变的影响比较小,适用点加载理论,而在扰度较大的情形下,压头的大小对石墨烯的形变状态的影响比较大,应该考虑球形压头大小对杨氏模量计算的影响。本文采用点加载模式对扰度较小时的数据组进行了分析以及采用球形压头加载模式对扰度较大时的数据组进行了分析,得出单层石墨烯的杨氏模量为1.00±0.05TPa。
2.分析了压头的半径的大小、圆膜尺寸的大小以及薄膜的预应力的不同对薄膜的杨氏模量计算值的影响,分析比较了采用加载作用力与薄膜扰度的关系为立方关系或者线性项加立方项的对应关系所获计算结果的异同。
3.研究了多层石墨烯的杨氏模量,提出了等效压头的概念。利用扰度较大时的数据组进行分析获得多层石墨烯的杨氏模量,得到了1-5层石墨烯的杨氏模量值为1.06TPa,0.82TPa,0.78TPa,0.91TPa,1.00TPa。模拟计算结果表明石墨烯的杨氏模量值对石墨烯的厚度具有一定的依赖作用,当石墨烯薄膜的厚度在3个原子层以下时,随着厚度的增加,杨氏模量会逐渐减小,当厚度大于3个原子层是,随着厚度的增加,杨氏模量又会逐渐的增大,并最终趋近于块体的杨氏模量值。
Graphene has broad application prospects due to its perfect optical, electrical,magnetic, and mechanical properties, which make it an excellent platform to design anovel class of advanced composites, impermeable membrane, electric transistors,semiconductor devices, nanosensors, and micro-nano electronic and mechanicalsystem, etc. However these excellent properties of graphene are related to itsmechanical properties which also determine the stability and reliability of the systemsequipped with graphene.
In this paper, during the molecular dynamics method (MD) and the employscircular membrane elastic theory, we have studied the Young’s modulus of freestanding circular membrane graphene. The main contents of this study are as follows:
1. According to the different stages of the deflection, we perform studiesrespectively. There are different elastic responding characters of different deflectionstages. In small deflection, the indenter has little effect on the film deformationcharacter and point indenter loading model is suitable for depicting the force loading;in larger deflection, the influence of the film deformation caused by indenter shouldnot be ignored, and spherical indenter loading model should be taken. From the smalland larger deflection data sets analysis, we determine the Young's modulus of themonolayer graphene which is about 1.0±0.05 TPa.
2. We have performed the analysis of the impact on the calculation results by theradius of the indenters, the size of the membrane, and the different pre-strain.Different results from cubic or polynomial force-deflection function for data setsfitting have also been discussed.
3. The young's modulus of multilayer graphene has been investigated. We proposeda concept of equivalent indenter. By using larger deflection data sets and employingspherical indenter loading model, the Young's modulus values of 1-5 layers graphene aredetermined, which are 1.06 TPa, 0.82 TPa, 0.78 TPa, 0.91 TPa, 1.00 TPa, respectively.The simulation results show that: the graphene Young’s modulus of graphene depends ontheir thickness. As the thickness is less than 3 atom layers, the modulus will decrease asthe thickness increases. However, when the thickness is more than 3 atom's layers themodulus will increase as the thickness increases, and equal to the bulk graphite modulusat last.
本文采用分子动力学(MD)方法利用圆膜弹性理论对独立式悬置圆膜石墨烯进行模拟计算获得石墨烯的杨氏模量。研究的主要内容如下:
1.按扰度的大小的不同采用分阶段研究的方法研究了单层石墨烯的杨氏模量。薄膜在扰度较小时和扰度较大时具有不同的弹性反应特征。在扰度较小的情况下,压头对薄膜形变的影响比较小,适用点加载理论,而在扰度较大的情形下,压头的大小对石墨烯的形变状态的影响比较大,应该考虑球形压头大小对杨氏模量计算的影响。本文采用点加载模式对扰度较小时的数据组进行了分析以及采用球形压头加载模式对扰度较大时的数据组进行了分析,得出单层石墨烯的杨氏模量为1.00±0.05TPa。
2.分析了压头的半径的大小、圆膜尺寸的大小以及薄膜的预应力的不同对薄膜的杨氏模量计算值的影响,分析比较了采用加载作用力与薄膜扰度的关系为立方关系或者线性项加立方项的对应关系所获计算结果的异同。
3.研究了多层石墨烯的杨氏模量,提出了等效压头的概念。利用扰度较大时的数据组进行分析获得多层石墨烯的杨氏模量,得到了1-5层石墨烯的杨氏模量值为1.06TPa,0.82TPa,0.78TPa,0.91TPa,1.00TPa。模拟计算结果表明石墨烯的杨氏模量值对石墨烯的厚度具有一定的依赖作用,当石墨烯薄膜的厚度在3个原子层以下时,随着厚度的增加,杨氏模量会逐渐减小,当厚度大于3个原子层是,随着厚度的增加,杨氏模量又会逐渐的增大,并最终趋近于块体的杨氏模量值。
Graphene has broad application prospects due to its perfect optical, electrical,magnetic, and mechanical properties, which make it an excellent platform to design anovel class of advanced composites, impermeable membrane, electric transistors,semiconductor devices, nanosensors, and micro-nano electronic and mechanicalsystem, etc. However these excellent properties of graphene are related to itsmechanical properties which also determine the stability and reliability of the systemsequipped with graphene.
In this paper, during the molecular dynamics method (MD) and the employscircular membrane elastic theory, we have studied the Young’s modulus of freestanding circular membrane graphene. The main contents of this study are as follows:
1. According to the different stages of the deflection, we perform studiesrespectively. There are different elastic responding characters of different deflectionstages. In small deflection, the indenter has little effect on the film deformationcharacter and point indenter loading model is suitable for depicting the force loading;in larger deflection, the influence of the film deformation caused by indenter shouldnot be ignored, and spherical indenter loading model should be taken. From the smalland larger deflection data sets analysis, we determine the Young's modulus of themonolayer graphene which is about 1.0±0.05 TPa.
2. We have performed the analysis of the impact on the calculation results by theradius of the indenters, the size of the membrane, and the different pre-strain.Different results from cubic or polynomial force-deflection function for data setsfitting have also been discussed.
3. The young's modulus of multilayer graphene has been investigated. We proposeda concept of equivalent indenter. By using larger deflection data sets and employingspherical indenter loading model, the Young's modulus values of 1-5 layers graphene aredetermined, which are 1.06 TPa, 0.82 TPa, 0.78 TPa, 0.91 TPa, 1.00 TPa, respectively.The simulation results show that: the graphene Young’s modulus of graphene depends ontheir thickness. As the thickness is less than 3 atom layers, the modulus will decrease asthe thickness increases. However, when the thickness is more than 3 atom's layers themodulus will increase as the thickness increases, and equal to the bulk graphite modulusat last.
引文
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[2] Peierls R. Quelques proprietes typiques des corpses solides [J]. Ann Inst Henri Poincare,1935, 5(3): 177-C222.
[3] Landau L. Lifshitz E: Statistical Physics, Part 1, 3rd edit: Pergamon Press [M], Oxford, NY;1980.
[4] Mermin N D. Crystalline order in two dimensions [J]. Phys Rev, 1968, 176(1): 250.
[5] Novoselov K S, Geim A K, Morozov S V et al. Electric Field Effect in Atomically ThinCarbon Films [J]. Science, 2004, 306(5696): 666-669.
[6] Geim A K, Novoselov K S. The rise of graphene [J]. Nature materials, 2007, 6(3): 183-191.
[7] Geim A K, Graphene: status and prospects [J]. Science, 2009, 324(5934): 1530.
[8] Novoselov K, Geim A, Morozov S et al. Two-dimensional gas of massless Dirac fermions ingraphene [J]. Nature, 2005, 438(7065): 197-200.
[9] Zhang Y, Tan Y W, Stormer H L et al. Experimental observation of the quantum Hall effectand Berry's phase in graphene [J]. Nature, 2005, 438(7065): 201-204.
[10] Brey L, Fertig H. Electronic states of graphene nanoribbons studied with the Dirac equation[J]. Phys Rev B, 2006, 73(23): 235411.
[11] Fujita M, Wakabayashi K, Nakada K et al. Peculiar localized state at zigzag graphite edge [J].J Phys Soc Jpn, 1996, 65(7): 1920-1923.
[12] Nakada K, Fujita M, Dresselhaus G et al. Edge state in graphene ribbons: Nanometer sizeeffect and edge shape dependence [J]. Phys Rev B, 1996, 54(24): 17954.
[13] Sasaki K, Murakami S, Saito R. Gauge field for edge state in graphene [J]. J Phys Soc Jpn,2006, 75(7): 074713.
[14] Sasaki K, Murakami S, Saito R. Stabilization mechanism of edge states in graphene [J]. Applphys lett, 2006, 88(11): 113110-113113.
[15] Wakabayashi K, Fujita M, Ajiki H et al. Electronic and magnetic properties of nanographiteribbons [J]. Phys Rev B, 1999, 59(12): 8271.
[16] Mao Y, Yuan J, Zhong J. Density functional calculation of transition metal adatom adsorptionon graphene [J]. J Phys: Condens Matter, 2008, 20: 115209.
[17] Mao Y, Zhong J. Structural, electronic and magnetic properties of manganese doping in theupper layer of bilayer graphene [J]. Nanotechnology, 2008, 19: 205708.
[18] Chen Y P, Xie Y E, Sun L et al. Asymmetric transport in asymmetric T-shaped graphenenanoribbons [J]. Appl phys lett, 2008, 93: 092104.
[19] Chen Y P, Xie Y E, Yan X H. Electron transport of L-shaped graphene nanoribbons [J]. J ApplPhys, 2008, 103: 063711.
[20] Xie Y E, Chen Y P, Zhong J X. Electron transport of folded graphene nanoribbons [J]. J ApplPhys, 2009, 106(10): 103714.
[21] Son Y W, Cohen M L, Louie S G. Half-metallic graphene nanoribbons [J]. Nature, 2006,444(7117): 347-349.
[22] Li X, Zhang G, Bai X et al. Highly conducting graphene sheets and Langmuir-Blodgett films[J]. Nat Nano, 2008, 3(9): 538-542.
[23] Geim A, Novoselov K. The rise of graphene [J]. Nature materials, 2007, 6(3): 183-191.
[24] Geim A. Graphene: status and prospects [J]. Science, 2009, 324(5934): 1530.
[25] Neto A H C, Guinea F, Peres N et al. The electronic properties of graphene [J]. Rev ModernPhys, 2009, 81(1): 109.
[26] Booth T, Blake P, Nair R et al. Macroscopic graphene membranes and their extraordinarystiffness [J]. Nano Lett, 2008, 8(8): 2442-2446.
[27] Schedin F, Geim A, Morozov S et al. Detection of individual gas molecules adsorbed ongraphene [J]. Nature materials, 2007, 6(9): 652-655.
[28] Lee C, Wei X, Kysar J et al. Measurement of the elastic properties and intrinsic strength ofmonolayer graphene [J]. Science, 2008, 321(5887): 385.
[29] Bunch J S, Verbridge S S, Alden J S et al. Impermeable atomic membranes from graphenesheets [J]. Nano letters, 2008, 8(8): 2458-2462.
[30] Frank I, Tanenbaum D, Van der Zande A et al. Mechanical properties of suspended graphenesheets [J]. J Vac Sci Technol B, 2007, 25: 2558.
[31] Colonna F, Los J H, Fasolino A et al. Properties of graphite at melting from multilayerthermodynamic integration [J]. Phys Rev B, 2009, 80(13): 134103.
[32] Neek-Amal M, Peeters F M, Nanoindentation of a circular sheet of bilayer graphene [J]. PhysRev B, 2010, 81(23): 235421.
[33] Van Lier G, Van Alsenoy C, Van Doren V et al. Ab initio study of the elastic properties ofsingle-walled carbon nanotubes and graphene [J]. Chem Phys Lett , 2000, 326(1-2): 181-185.
[34] Kudin K, Scuseria G, Yakobson B. C2F, BN, and C nanoshell elasticity from ab initiocomputations [J]. Phys Rev B, 2001, 64(23): 235406.
[35] Natsuki T, Tantrakarn K, Endo M. Prediction of elastic properties for single-walled carbonnanotubes [J]. Carbon, 2004, 42(1): 39-45.
[36] Li C, Chou T. A structural mechanics approach for the analysis of carbon nanotubes [J]. Int JSolids Structures, 2003, 40(10): 2487-2499.
[37] Sakhaee-Pour A. Elastic buckling of single-layered graphene sheet [J].Comput Mater Sci,2009, 45(2): 266-270.
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