原子力显微镜微悬臂梁刚度测量研究
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摘要
随着原子力显微镜(Atomic Force Microscopy,AFM)的相关研究工作逐渐深入,人们已经不仅仅满足于AFM的形貌测量功能,如何利用AFM开展微小力测量逐渐成为AFM应用研究的热点之一,并使得AFM成为了纳米摩擦学、生物力学等领域研究的重要工具。AFM微悬臂梁的刚度参数是连接AFM输出信号和针尖上作用力大小的重要桥梁,因此对微悬臂梁刚度参数的测量是实现基于AFM的微小力测量的重要前提。本文根据微悬臂梁的特点,结合AFM力学测量中面临的实际问题,进行了AFM微悬臂梁刚度动态测量技术的研究。
建立了矩形和V形微悬臂梁的频率响应近似理论模型和有限元分析模型,并针对一组典型的矩形微悬臂梁和V形微悬臂梁进行了仿真计算。在此基础上,建立了微悬臂梁杨氏模量与其离面弯曲固有频率之间的关系模型,并通过有限元仿真工具进行了验证。
大部分AFM力学测量实验在空气或水等流体环境中完成。实验表明,微悬臂梁在流体中发生振动时,其共振频率会受到流体阻尼的影响而发生漂移,为了定量的研究微悬臂梁频率响应特性,进而为微悬臂梁杨氏模量和刚度动态测量提供必要的理论基础,本文详细的研究了流体环境对微悬臂梁频率响应的影响。首先提出了用于分析微悬臂梁在流体中频率响应问题的简化一维法,该方法可以将微悬臂梁在流体中振动问题简化为弹簧质量系统的阻尼振动问题,在此基础上推导了微悬臂梁在流体和真空中的固有频率关系公式。鉴于简化一维法的计算结果与实验结果存在较大误差,本文讨论了微悬臂梁的尺寸效应和流固耦合振动现象。为了进一步研究流体环境对微悬臂梁频率响应的影响,提出了用于分析微悬臂梁流固耦合振动问题的等效气弹模型法和计算流体力学法,并针对矩形和V形微悬臂梁进行了仿真分析。仿真结果表明,基于计算流体力学法的微悬臂梁频率响应分析模型可以较真实的反映微悬臂梁在空气环境中的振动问题。
基于本文建立的微悬臂梁流固耦合振动模型,提出了微悬臂梁刚度动态测试方法。此方法在操作时分三步:第一步测量微悬臂梁在空气中的离面弯曲共振频率;第二步利用数值方法计算相应于不同杨氏模量的微悬臂梁离面弯曲共振频率序列,与实验结果最吻合的计算值所对应的杨氏模量为待测杨氏模量。第三步根据第二步中测量的微悬臂梁杨氏模量计算微悬臂梁的刚度。基于此方法,测量了NSC14/No Al矩形微悬臂梁、NSC11/No Al-large和NSC11/NoAl-small V形微悬臂梁的杨氏模量和刚度。
微悬臂梁的理论刚度与其工作刚度之间存在一定的偏差,如果忽略了这个偏差,会给基于AFM的微小力定量测量带来较大误差。为了提高AFM力学测量的精度,分别研究了微悬臂梁安装倾斜角度、针尖安装位置、接触刚度和面内形变对微悬臂梁等效刚度的影响,建立了微悬臂梁等效刚度计算公式,并针对本文使用的三种微悬臂梁进行了计算。计算结果表面微悬臂梁安装倾斜角度、针尖安装位置、接触刚度和面内形变对微悬臂梁等效刚度具有较大影响。
提出了微悬臂梁等效刚度校正方法。特别研究了微悬臂梁安装倾斜角度、接触刚度对于AFM纳米摩擦学实验的影响,在此基础上,研究了接触刚度对AFM径向力校准的影响,并提出了校正方法,实验证明使用该校正方法可有效避免AFM径向力校准系数受到法向力影响。
Atomic force microscopy (AFM) is widely used as a tool for detection of small forces, and becomes an important measuring instrument in the research field of nanotribology and bio-mechanics. The spring constant is an important bridge in converting AFM signals to forces loaded on the tips, and it is of great importance to develop a reliable and easy method for cantilevers' spring constant measurement. This dissertation is focusing on dynamic method for determination of spring constants of AFM cantilevers.
A simplified theoretical modeling as well as a FEA method is developed for researching the frequency response of rectangular and v-shaped cantilevers in vacuum. Simulation on some common used rectangular and v-shaped cantilevers has been down for validation. Modeling for interpreting the relationship between young's modulus and the base frequency of out-of-plane bending vibration is well developed, and validated through a FEA method.
It is well known that the frequency response of an elastic beam is strongly dependent on the fluid in which it is immersed. The frequency response of cantilevers immersed in fluid is detailed discussed in the dissertation for the convenience of research on quantitative analysis for dynamic properties of microcantilevers and the preparation for theoretical fundaments of spring constant determination through dynamic tests methods. A simplified one-dimension modeling for frequency response of microcantilevers immersed in fluid is developed to ease the problem as the damping vibration of a spring-mass system at first. Consequently, the formula indicating the ration between the natural frequency of cantilevers in vacuum and fluid is presented. Because the simulation based on the simplified method show a evident difference against experimental experiences, the fluid-structure interaction (FSI) problem of microcantilevers immersed in fluid is discussed lately. Two methods are presented for solving the FSI problem of microcantilevers, which are air-spring modeling method and computational fluid dynamic (CFD) method, comparative simulation have been down on a rectangular cantilever and a v-shaped cantilever, which indicates a good agreement with the experience.
A dynamic method for determination of cantilever spring constants is presented based on the CFD modeling presented in this dissertation, which involves three steps. The first step is measuring the resonance frequency of a micro cantilever immersed in air. The second step is calculation of resonance frequencies of the microcantilever with different young’s modulus, and the young’s modulus can be determined after comparison with the measured values. The third step is taking the determined young’s modulus to the FEA model of the microcantilever to determine its spring constant. The spring constants of a NSC14/No Al, a NSC11/No Al-large cantilever and a NSC11/No Al-small cantilever have been measured based in the dynamic method.
The effective spring constant of microcantilevers are differing from theoretical ones, and terrible errors could happen with out consideration of it. The effect of cantilever tilt angle, in-plane deformation, contact stiffness and tip position on effective spring constants of cantilevers are detailed discussed. Theoretical formulae have been presented to correct the spring constants. Related experiments on the cantilevers used in this dissertation have been done for validation. The effect of those factors on AFM based nanotribology measurements are discussed, and a correction method in lateral force calibration is presented, which is of great value to the users and designers of AFM.
建立了矩形和V形微悬臂梁的频率响应近似理论模型和有限元分析模型,并针对一组典型的矩形微悬臂梁和V形微悬臂梁进行了仿真计算。在此基础上,建立了微悬臂梁杨氏模量与其离面弯曲固有频率之间的关系模型,并通过有限元仿真工具进行了验证。
大部分AFM力学测量实验在空气或水等流体环境中完成。实验表明,微悬臂梁在流体中发生振动时,其共振频率会受到流体阻尼的影响而发生漂移,为了定量的研究微悬臂梁频率响应特性,进而为微悬臂梁杨氏模量和刚度动态测量提供必要的理论基础,本文详细的研究了流体环境对微悬臂梁频率响应的影响。首先提出了用于分析微悬臂梁在流体中频率响应问题的简化一维法,该方法可以将微悬臂梁在流体中振动问题简化为弹簧质量系统的阻尼振动问题,在此基础上推导了微悬臂梁在流体和真空中的固有频率关系公式。鉴于简化一维法的计算结果与实验结果存在较大误差,本文讨论了微悬臂梁的尺寸效应和流固耦合振动现象。为了进一步研究流体环境对微悬臂梁频率响应的影响,提出了用于分析微悬臂梁流固耦合振动问题的等效气弹模型法和计算流体力学法,并针对矩形和V形微悬臂梁进行了仿真分析。仿真结果表明,基于计算流体力学法的微悬臂梁频率响应分析模型可以较真实的反映微悬臂梁在空气环境中的振动问题。
基于本文建立的微悬臂梁流固耦合振动模型,提出了微悬臂梁刚度动态测试方法。此方法在操作时分三步:第一步测量微悬臂梁在空气中的离面弯曲共振频率;第二步利用数值方法计算相应于不同杨氏模量的微悬臂梁离面弯曲共振频率序列,与实验结果最吻合的计算值所对应的杨氏模量为待测杨氏模量。第三步根据第二步中测量的微悬臂梁杨氏模量计算微悬臂梁的刚度。基于此方法,测量了NSC14/No Al矩形微悬臂梁、NSC11/No Al-large和NSC11/NoAl-small V形微悬臂梁的杨氏模量和刚度。
微悬臂梁的理论刚度与其工作刚度之间存在一定的偏差,如果忽略了这个偏差,会给基于AFM的微小力定量测量带来较大误差。为了提高AFM力学测量的精度,分别研究了微悬臂梁安装倾斜角度、针尖安装位置、接触刚度和面内形变对微悬臂梁等效刚度的影响,建立了微悬臂梁等效刚度计算公式,并针对本文使用的三种微悬臂梁进行了计算。计算结果表面微悬臂梁安装倾斜角度、针尖安装位置、接触刚度和面内形变对微悬臂梁等效刚度具有较大影响。
提出了微悬臂梁等效刚度校正方法。特别研究了微悬臂梁安装倾斜角度、接触刚度对于AFM纳米摩擦学实验的影响,在此基础上,研究了接触刚度对AFM径向力校准的影响,并提出了校正方法,实验证明使用该校正方法可有效避免AFM径向力校准系数受到法向力影响。
Atomic force microscopy (AFM) is widely used as a tool for detection of small forces, and becomes an important measuring instrument in the research field of nanotribology and bio-mechanics. The spring constant is an important bridge in converting AFM signals to forces loaded on the tips, and it is of great importance to develop a reliable and easy method for cantilevers' spring constant measurement. This dissertation is focusing on dynamic method for determination of spring constants of AFM cantilevers.
A simplified theoretical modeling as well as a FEA method is developed for researching the frequency response of rectangular and v-shaped cantilevers in vacuum. Simulation on some common used rectangular and v-shaped cantilevers has been down for validation. Modeling for interpreting the relationship between young's modulus and the base frequency of out-of-plane bending vibration is well developed, and validated through a FEA method.
It is well known that the frequency response of an elastic beam is strongly dependent on the fluid in which it is immersed. The frequency response of cantilevers immersed in fluid is detailed discussed in the dissertation for the convenience of research on quantitative analysis for dynamic properties of microcantilevers and the preparation for theoretical fundaments of spring constant determination through dynamic tests methods. A simplified one-dimension modeling for frequency response of microcantilevers immersed in fluid is developed to ease the problem as the damping vibration of a spring-mass system at first. Consequently, the formula indicating the ration between the natural frequency of cantilevers in vacuum and fluid is presented. Because the simulation based on the simplified method show a evident difference against experimental experiences, the fluid-structure interaction (FSI) problem of microcantilevers immersed in fluid is discussed lately. Two methods are presented for solving the FSI problem of microcantilevers, which are air-spring modeling method and computational fluid dynamic (CFD) method, comparative simulation have been down on a rectangular cantilever and a v-shaped cantilever, which indicates a good agreement with the experience.
A dynamic method for determination of cantilever spring constants is presented based on the CFD modeling presented in this dissertation, which involves three steps. The first step is measuring the resonance frequency of a micro cantilever immersed in air. The second step is calculation of resonance frequencies of the microcantilever with different young’s modulus, and the young’s modulus can be determined after comparison with the measured values. The third step is taking the determined young’s modulus to the FEA model of the microcantilever to determine its spring constant. The spring constants of a NSC14/No Al, a NSC11/No Al-large cantilever and a NSC11/No Al-small cantilever have been measured based in the dynamic method.
The effective spring constant of microcantilevers are differing from theoretical ones, and terrible errors could happen with out consideration of it. The effect of cantilever tilt angle, in-plane deformation, contact stiffness and tip position on effective spring constants of cantilevers are detailed discussed. Theoretical formulae have been presented to correct the spring constants. Related experiments on the cantilevers used in this dissertation have been done for validation. The effect of those factors on AFM based nanotribology measurements are discussed, and a correction method in lateral force calibration is presented, which is of great value to the users and designers of AFM.
引文
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73. T. NOMURA. ALE finite element computations of fluid~structure interaction problems. Computer methods in applied mechanics and engineering. 1994, 112(1~4):291~308
74. S. R. Idelsohn, E. O?ate, F. Del Pin. A Lagrangian meshless finite element method applied to fluid~structure interaction problems. Computers & Structures. 2003, 81(8~11):655~671
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76.沈世钊,武岳.膜结构风振响应中的流固耦合效应研究进展.建筑科学与工程学报. 2006, 23(1):1~9
77. M. Souli, A. Ouahsine and L. Lewin. ALE formulation for fluid structure interaction problems. Computaton Methods in Applied Mechanics and Engineering. 2000, 190:559~675
78.邬海军,郭鹏程,廖伟丽等. ANSYS在水轮机部件流固耦合振动分析中的应用水电能源科学,2004,22(4):64~66
79.张天林,郭哲颖,张青川等.微梁传感器弹性模量的测试[J].中国科学技术大学学报, 2006,36(5):567~572
80. R. B. King. Elastic analysis of some punch problems for a layered medium. International Journal of Solids Structures. 1987, 23:1657~1664
81. Y. T. Cheng, C. M. Cheng. What is indentation hardness? Surface and Coatings Technology. 2000, 417:133~134
82. J. L. Hay, W. C. Oliver, A. Bolshakov, G.M. Pharr. Using the ratio of loading slope and elastic stiffness to predict pile-up and constraint factor during indentation. In: Proceedings of Materials Research Society Symposium, Pittsburgh. 1998
83. J. S. Field, M. V. Swain. A simple predictive model for spherical indentation. Journal of Materials Research. 1993, 8:297~306
84. W. C. Oliver, G. M. Pharr. An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. Journal of Materials Research. 1992, 7:1564~1583
85. H. Lee, S. Mall and PE. Kladitis. Nanoindentation technique for characterizing cantilever beam style RF microelectromechanical systems. Journal of Micromechanics and Micro-engineering. 2005, 15:1230~1235
86. J.A. Ruan, B. Bhushan. Atomic~scale Friction Measurements using Friction Force Microscopy : Part I. ASME J. Tribol. 1994,116:378
87. E. Liu, B. Blanpain and J. P. Celis. Calibration procedures for frictional measurements with a lateral force microscope. Wear. 1996, 192:141~150
88. R. W. Carpick, N. Agrait, D. F. Ogletree and M. Salmeron. Measurement of interfacial shear (friction) with an ultrahigh vacuum atomic force microscope.J. Vac. Sci. Technol. B. 1996,14(2):1289~1295
89. S. Sundararajan, B. Bhushan. Development of AFM~based techniques tomeasure mechanical properties of nanoscale structures. Sensors & Actuators: A, Physical. 2002, 101(3):338~351
90. M. Evstigneev, A. Schirmeisen, L Jansen, H. Fuchs. Force Dependence of Transition Rates in Atomic Friction. Physical Review Letters. 2006, 97(24):240601.1~240601.4
91. N. Morel, Ph. Tordjeman, M. Ramonda. Cantilever calibration for nanofriction experiments with atomic force microscope. Journal of Physics D: Applied Physics. 2005, 86(16): 163103~163103.3
92. P. J. Cumpson, P. Zhdan, J. Hedley. Accurate force measurement in the atomic force microscope: a microfabricated array of reference springs for easy cantilever calibration. Nanotechnology. 2003, 14:918~924
93. P. J. Cumpson, J. Hedley. Accurate force measurement in the atomic force microscope: a microfabricated array of reference springs for easy cantilever calibration. Nanotechnology. 2003, 14:1279~1288
94. P. J. Cumpson, P. Zhdan, J. Hedley. Calibration of AFM cantilever stiffness: a microfabricated array of reflective springs. Ultramicroscopy. 2004, 100(3~4):241~251
95. P. J. Cumpson, C. A. Clifford, J. Hedley. Quantitative analytical atomic force microscopy: a cantilever reference device for easy and accurate AFM spring~constant calibration. Meas. Sci. Technol.. 2003, 15:1337~1346
96. P. J. Cumpson, J. Hedley. C. A. Clifford. Microelectromechanical device for lateral force calibration in the atomic force microscope: Lateral electrical nanobalance. Journal of Vacuum Science & Technology B: Microelectronics and Nanometer Structures. 2005, 23(5):1992~1997
97. P. J. Cumpson, J. Hedley. C. A. Clifford, Xinyong Chen, S. Allen. Microelectromechanical system device for calibration of atomic force microscope cantilever spring constants between 0.01 and 4 N/m. Journal of Vacuum Science & Technology A: Vacuum, Surfaces, and Films. 2004, 22(4):1444~1449
98. Laurent Nony, Rodolphe Boisgard, Jean-Pierre Aimé. Nonlinear dynamical properties of an oscillating tip~cantilever system in the tapping mode. Journal of Chemical Physics. 1999, 111(4): 1615~1627
99. H. X. You, J. M. Lau, S. Zhang, L. Yu L. Atomic force microscopy imaging ofliving cells: a preliminary study of the disruptive effect of the cantilever tip on cell morphology. Ultramicroscopy. 2000, 82(1~4):297~305
100. Naruo Sasaki and Masaru Tsukada. Theory for the effect of the tip~surface interaction potential on atomic resolution in forced vibration system of noncontact AFM. Applied Surface Science. 1999, 140(34):339~343
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