分离式拟质点方法的研究及纳米压痕多尺度模拟
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摘要
随着近几十年大规模集成电路技术和纳米技术的迅猛发展,特别是当电路的特征宽度降低到纳米级之后,通过对各类电子部件压痕特性的分析以揭示其纳米尺度下的力学特性已成为当今应用研究的关注点之一。另一方面,诸如金在室温下可变为液态、绝缘体硅能变成导体等这些新颖的材料特性也强烈地激发着人们对纳米尺度下各类物理现象进行不断探索的热情。而所有这些问题的解决都离不开多尺度分析方法的发展与完善。尤其是跨原子/连续介质多尺度分析方法的研究在其中占据着重要地位。但目前常用的多尺度分析方法在连续介质区域一般采用有限元方法来加以模拟计算,这一处理方法容易在原子尺度和有限元尺度的边界处产生如“鬼力”等各类虚假的物理现象,从而导致两个尺度间的物理量并不能光滑无碍地相互传递。同时在有限元的某些区域,有限元节点间距几乎与原子间距相等。这又将消耗大量的存储空间和计算时间。最近几年,有研究者将连续介质尺度内的模型离散化,采用质点系统来代替以往的有限元系统,进而发展出一类被称之为原子/质点多尺度模拟方法。该类方法以其独特的几何构造和清晰的物理概念吸引着人们对其不断进行探索和完善。本论文在已有的原子/质点多尺度分析方法的基础上,从计算质点位置信息和受力信息的角度作为切入点,围绕原子与质点多尺度模拟方法的建立和应用等问题开展了以下几方面的研究：
1.根据原子/质点模型的基本原理,建立了与面心立方晶胞相对应的新型二级拟质点三维空间几何模型,使得拟质点的空间位置分布更加符合材料的实际特点。并推导出该拟质点几何模型与面心立方晶胞之间的定量数值关系。同时在原子系统和拟质点系统的界面处分别构造出虚拟的原子区域和拟质点区域,从而将自然边界条件引入到分离式拟质点多尺度分析模型中。因此可以保证系统的位移和应力等物理量能无碍地在原子系统区和拟质点系统区之间互相传递。
2.由平均加速度方法求解出此时的拟质点系统势能函数的参数,并将其应用于对拟质点的加速度、速度和位置等信息的计算中；基于能量守恒定律求解出拟质点系统另一组的势能函数参数,并将其应用于对各个拟质点系统受力信息的计算中。进而绕过牛顿第二定律,使拟质点的位置和受力等物理量与实际原子系统的物理信息保持一致。
3.以单晶铜纳米丝模型为例,对比分析分离式拟质点多尺度方法、广义质点多尺度方法以及分子动力学方法在不同加载速率条件下模拟实验的结果,以验证分离式拟质点多尺度分析方法的正确性。
4.采用分离式拟质点多尺度分析方法对含表面缺陷的单晶铜薄膜纳米压痕进行了模拟研究。分别讨论了缺陷与压头的距离取四种不同数值时,金属薄膜受压荷载与压头位移之间的变化规律。通过对模型x位移分布图以及Mises局部剪应变不变量分布图的分析比较,揭示了缺陷对材料内部原子微观结构的影响规律。并进一步计算和讨论了表面缺陷对材料内部Peierls应力的影响效果。
5.应用本论文提出的新型多尺度分析方法分别对金属铜、铝和银的纳米压痕尺寸效应进行了模拟分析。根据对模拟得到的三种金属材料在四种试件长度和四种压头宽度条件下的荷载与位移响应曲线的分析与比较,揭示出材料各个物理参数对其内部位错形核的作用规律。绘制出各种情况下压头下方材料Mises局部剪切应变不变量分布图和沿x方向位移的分布图。从而直观地体现出压头加载过程中,材料内部微观结构的变化细节。将模拟计算得到的三种金属材料的硬度值与文献中的结果进行比较,证明了该多尺度分析方法的合理性。以金属铝薄膜纳米压痕为例,详细地讨论了位错形核时所对应的临界荷载与压头宽度的关系,并将计算得到的数据与理论值以及QC方法得到的模拟值比较,进一步验证了该分离式拟质点多尺度分析方法的正确性。
综上所述,本文对一种新型的跨原子/质点的多尺度方法进行了理论研究。建立了该多尺度分析方法的几何模型及相关的动力学方程,并将其应用于各种金属薄膜的纳米压痕分析中,并希望本论文的工作能为纳米工程领域内的多尺度研究提供参考和帮助。
In recent decades, the very large-scale integrated circuits (VLSI) technology and nanotechnology have developed rapidly. Especially when the characteristic width of VLSI reduces to nanometer level, the study on indentation, which reveals the nano-scale mechanical properties in various types of electronic components, has become one focus of application researches. Furthermore, some new material properties, such as gold may become liquid at room temperature; insulator silicon may become the conductor, also strongly arouses the enthusiasm of scholars in exploring the surprising nano-scale phenomena. Yet the solving of the all problems is not separable in the development and perfection of multiscale methods. In especial, the multiscale method of cross atoms and continuum plays an important role in the multiscale analysis. However, the usual multiscale methods employ the finite element method for simulation in continuum region. The approach could easily produce some false phenomena (e.g. ghost force) at the boundary between atomic model and finite element model. It may case that the physical quantities do not pass through boundary smoothly. Meanwhile, the treatment that the distance of finite element nodes is equal to the one of adjacent atoms would consume a large amount of storage space and computation time. In the last years, researchers have made the model discretization in continuum scale, replaced the finite element system with particle system, and developed a new method, which is called as atomic/particle multiscale method. The kind of method that has its unique geometric construction and clear physical concept attracts researchers to develop and improve it continuously. In the present work, based on previous atomic/particle multiscale analysis methods, the calculation for quasi-particle position and quasi-particle interaction would be the basic breakthrough point. And the following aspects, focused on the establishment and applications of atomic/quasi-particle method, have been researched in detail.
1. According to the basic principles of atom/particle model, the new three-dimensional geometric model of second-scale quasi-particles has been established in the correspondence of face-centered cubic crystal lattice of atom system so that the spatial distribution of quasi-particles could be more in line with the actual characteristics of the material. The quantitative relationship between quasi-particle geometry model and face-centered cubic crystal lattice has been proposed. Meanwhile, the construction of virtual quasi-particle and virtual atomic regions at both sides of atomic system and quasi-particle system introduces natural boundary conditions into the separate quasi-particle multiscale method to ensure the displacement, stress or other physical quantities could pass thought freely between atomic area and quasi-particle area.
2. In the quasi-particle system, the parameters of potential function, which are solved by average acceleration method, are applied to calculate the quasi-particle acceleration, velocity and position. Furthermore, the parameters of other potential function are obtained by the law of conservation of energy. And they are used to calculate the interaction of quasi-particles. In above process, the position and force of quasi-particle system keep in step with actual physical information of atom system by avoiding introducing Newton's second law directly,
3. To take the single-crystal copper nanowire for example, the simulation results of the separate quasi-particle method, generalized particle method and molecular dynamics method under different loading rates have been compared and analyzed to validate the correct for the separate quasi-particle method.
4. The single crystal copper thin films with surface defects have been simulated by using the separate quasi-particle method. The variations between the load on thin film and the displacement of indenter are discussed when the distance between defect and indenter takes four different values. Meanwhile, by comparing to the x-direction displacement distribution and Mises local shear strain invariant distribution, the effects of defect on the material atomic system microstructure have been systematically revealed. And the influence of surface defect on the Peierls stress has been calculated and discussed.
5. The size effect of copper, aluminum and silver nanoindentation has been studied by adopting the new multiscale method. According to the analysis and comparison of load-displacement cures for the three materials under the four specimens and four indenter widths, the effects of various physical parameters on the dislocation nucleation have been discussed. The Mises local shear strain invariant distribution and x-direction displacement distribution have been plotted to directly reflect the material details in the process of loading. Comparing the hardness values of the three materials with the result of the literature, the multiscale method has been proved correct. As an example, the nanoindentation at aluminum film is simulated in the discussion of relationship between critical load of dislocation nucleation and indenter width. The separate quasi-particle multiscale method is further proved to be correct with the comparison of results between the separate quasi-particle multiscale method, theoretical method and the QC method.
In conclusion, a new multiscale method cross atom/particle have been proposed and studied systematically. The geometric model and dynamic equations for this method have been obtained and applied in the analysis of nanoindentation of some metal films, which hopefully would provide useful reference to the nanotechnology research and engineering problems.
1.根据原子/质点模型的基本原理,建立了与面心立方晶胞相对应的新型二级拟质点三维空间几何模型,使得拟质点的空间位置分布更加符合材料的实际特点。并推导出该拟质点几何模型与面心立方晶胞之间的定量数值关系。同时在原子系统和拟质点系统的界面处分别构造出虚拟的原子区域和拟质点区域,从而将自然边界条件引入到分离式拟质点多尺度分析模型中。因此可以保证系统的位移和应力等物理量能无碍地在原子系统区和拟质点系统区之间互相传递。
2.由平均加速度方法求解出此时的拟质点系统势能函数的参数,并将其应用于对拟质点的加速度、速度和位置等信息的计算中；基于能量守恒定律求解出拟质点系统另一组的势能函数参数,并将其应用于对各个拟质点系统受力信息的计算中。进而绕过牛顿第二定律,使拟质点的位置和受力等物理量与实际原子系统的物理信息保持一致。
3.以单晶铜纳米丝模型为例,对比分析分离式拟质点多尺度方法、广义质点多尺度方法以及分子动力学方法在不同加载速率条件下模拟实验的结果,以验证分离式拟质点多尺度分析方法的正确性。
4.采用分离式拟质点多尺度分析方法对含表面缺陷的单晶铜薄膜纳米压痕进行了模拟研究。分别讨论了缺陷与压头的距离取四种不同数值时,金属薄膜受压荷载与压头位移之间的变化规律。通过对模型x位移分布图以及Mises局部剪应变不变量分布图的分析比较,揭示了缺陷对材料内部原子微观结构的影响规律。并进一步计算和讨论了表面缺陷对材料内部Peierls应力的影响效果。
5.应用本论文提出的新型多尺度分析方法分别对金属铜、铝和银的纳米压痕尺寸效应进行了模拟分析。根据对模拟得到的三种金属材料在四种试件长度和四种压头宽度条件下的荷载与位移响应曲线的分析与比较,揭示出材料各个物理参数对其内部位错形核的作用规律。绘制出各种情况下压头下方材料Mises局部剪切应变不变量分布图和沿x方向位移的分布图。从而直观地体现出压头加载过程中,材料内部微观结构的变化细节。将模拟计算得到的三种金属材料的硬度值与文献中的结果进行比较,证明了该多尺度分析方法的合理性。以金属铝薄膜纳米压痕为例,详细地讨论了位错形核时所对应的临界荷载与压头宽度的关系,并将计算得到的数据与理论值以及QC方法得到的模拟值比较,进一步验证了该分离式拟质点多尺度分析方法的正确性。
综上所述,本文对一种新型的跨原子/质点的多尺度方法进行了理论研究。建立了该多尺度分析方法的几何模型及相关的动力学方程,并将其应用于各种金属薄膜的纳米压痕分析中,并希望本论文的工作能为纳米工程领域内的多尺度研究提供参考和帮助。
In recent decades, the very large-scale integrated circuits (VLSI) technology and nanotechnology have developed rapidly. Especially when the characteristic width of VLSI reduces to nanometer level, the study on indentation, which reveals the nano-scale mechanical properties in various types of electronic components, has become one focus of application researches. Furthermore, some new material properties, such as gold may become liquid at room temperature; insulator silicon may become the conductor, also strongly arouses the enthusiasm of scholars in exploring the surprising nano-scale phenomena. Yet the solving of the all problems is not separable in the development and perfection of multiscale methods. In especial, the multiscale method of cross atoms and continuum plays an important role in the multiscale analysis. However, the usual multiscale methods employ the finite element method for simulation in continuum region. The approach could easily produce some false phenomena (e.g. ghost force) at the boundary between atomic model and finite element model. It may case that the physical quantities do not pass through boundary smoothly. Meanwhile, the treatment that the distance of finite element nodes is equal to the one of adjacent atoms would consume a large amount of storage space and computation time. In the last years, researchers have made the model discretization in continuum scale, replaced the finite element system with particle system, and developed a new method, which is called as atomic/particle multiscale method. The kind of method that has its unique geometric construction and clear physical concept attracts researchers to develop and improve it continuously. In the present work, based on previous atomic/particle multiscale analysis methods, the calculation for quasi-particle position and quasi-particle interaction would be the basic breakthrough point. And the following aspects, focused on the establishment and applications of atomic/quasi-particle method, have been researched in detail.
1. According to the basic principles of atom/particle model, the new three-dimensional geometric model of second-scale quasi-particles has been established in the correspondence of face-centered cubic crystal lattice of atom system so that the spatial distribution of quasi-particles could be more in line with the actual characteristics of the material. The quantitative relationship between quasi-particle geometry model and face-centered cubic crystal lattice has been proposed. Meanwhile, the construction of virtual quasi-particle and virtual atomic regions at both sides of atomic system and quasi-particle system introduces natural boundary conditions into the separate quasi-particle multiscale method to ensure the displacement, stress or other physical quantities could pass thought freely between atomic area and quasi-particle area.
2. In the quasi-particle system, the parameters of potential function, which are solved by average acceleration method, are applied to calculate the quasi-particle acceleration, velocity and position. Furthermore, the parameters of other potential function are obtained by the law of conservation of energy. And they are used to calculate the interaction of quasi-particles. In above process, the position and force of quasi-particle system keep in step with actual physical information of atom system by avoiding introducing Newton's second law directly,
3. To take the single-crystal copper nanowire for example, the simulation results of the separate quasi-particle method, generalized particle method and molecular dynamics method under different loading rates have been compared and analyzed to validate the correct for the separate quasi-particle method.
4. The single crystal copper thin films with surface defects have been simulated by using the separate quasi-particle method. The variations between the load on thin film and the displacement of indenter are discussed when the distance between defect and indenter takes four different values. Meanwhile, by comparing to the x-direction displacement distribution and Mises local shear strain invariant distribution, the effects of defect on the material atomic system microstructure have been systematically revealed. And the influence of surface defect on the Peierls stress has been calculated and discussed.
5. The size effect of copper, aluminum and silver nanoindentation has been studied by adopting the new multiscale method. According to the analysis and comparison of load-displacement cures for the three materials under the four specimens and four indenter widths, the effects of various physical parameters on the dislocation nucleation have been discussed. The Mises local shear strain invariant distribution and x-direction displacement distribution have been plotted to directly reflect the material details in the process of loading. Comparing the hardness values of the three materials with the result of the literature, the multiscale method has been proved correct. As an example, the nanoindentation at aluminum film is simulated in the discussion of relationship between critical load of dislocation nucleation and indenter width. The separate quasi-particle multiscale method is further proved to be correct with the comparison of results between the separate quasi-particle multiscale method, theoretical method and the QC method.
In conclusion, a new multiscale method cross atom/particle have been proposed and studied systematically. The geometric model and dynamic equations for this method have been obtained and applied in the analysis of nanoindentation of some metal films, which hopefully would provide useful reference to the nanotechnology research and engineering problems.
引文
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