Flexural vibrations of double tapered atomic force microscope cantilevers studied by considering the contact position and using the differential quadrature method

The resonant frequency of flexural vibrations for a double tapered atomic force microscope (AFM) cantilever has been investigated by using the Timoshenko beam theory. In this paper, the effects of various parameters on the dimensionless frequency of vibrations of the AFM cantilever have been studied. The differential quadrature method (DQM) is employed to solve the nonlinear differential equations of motion. The results show that the resonant frequency decreases when the Timoshenko beam parameter or the cantilever thickness increases, and high-order modes are more sensitive to it. The first frequency is sensitive only in the lower range of contact stiffness, but the higher-order modes are sensitive to the contact stiffness in a larger range. Increasing the tip height increases the sensitivity of the vibrational modes in a limited range of normal contact stiffness. Furthermore, with increasing the breadth taper ratio, the frequency increases. The DQM results are compared with the exact solution for a rectangular AFM cantilever.     

Investigation of Near-Surface Mechanical Properties of Materials Using Atomic Force Microscopy

As ultra-thin films or small-scale structures become widely used in electronics and biology, knowledge concerning their near-surface mechanical properties of the materials is increasingly important. Atomic force microscopy (AFM) is employed to determine near-surface elastic modulus via force-penetration curves acquired during indentation. Samples include polydimethylsiloxane (PDMS), parylene, mica, and single-crystal silicon, and indentations are performed with single-crystal silicon and silicon nitride AFM tips. An analysis algorithm based on the secant modulus method is proposed to extract the true penetration curves from the experimental displacement curves. The penetration data is then analyzed in terms of Hertzian model to estimate the elastic modulus. Three concerns in applying nanoscale AFM indentation to the measurement of the elastic modulus of an ultra-thin material are addressed. First, the effect of the lateral force caused by the inclined angle of the cantilevered probe is investigated theoretically and by numerical simulation. A second concern is local plastic deformation induced by a sharp probe tip. In this case, numerical results show a relatively small effect on the force-penetration curves if the plastic deformation is limited to the central area below the probe tip. The deviation of the elastic-plastic simulation from the elastic estimation depends on the yield strength of the material. Finally, the effect of stiffness matching between the AFM probe and the sample is a key issue that is studied numerically, and appropriate stiffness matching criteria are suggested.     

轻敲模式下 AFM 动力学模型及能量耗散机理研究

轻敲模式下探针从远离到间歇性接触样品表面,是一个连续的能量耗散过程.针对该连续过程的能量耗散机理研究仅零星存在于各个文献之中,对于连续过程中各个阶段的能量耗散机理也没有一个系统的解释和实验验证.本文提出了新的位移激励下原子力显微镜探针-样品系统简化模型并得到了一维振子系统等效阻尼的计算方法,并通过该方法计算了探针在远离样品表面时的空气黏性阻尼和靠近样品时的空气压膜阻尼,分析了探针从远离样品到间歇性接触样品表面这一过程中的环境耗散机理变化,得到了原子力显微镜系统理论品质因数与探针工作位置的关系曲线;在此基础上设计了轻敲模式下的微悬臂梁扫频实验,得到了系统实验品质因数与探针工作位置的关系曲线,进而验证了理论模型的准确性. 本文通过对轻敲模式下AFM环境耗散机理进行理论分析和实验验证,希望可以对轻敲模式下AFM动力学特性及其阻尼作用机理有更近一步的认识,同时对微纳米机电系统 (MEMS/NEMS) 能量耗散机理的研究提供理论参考和实验方法.      

STUDY ON A DYNAMICS MODEL OF TAPPING MODE AFM AND ENERGY DISSIPATION MECHANISM$^{\bf 1)}$

轻敲模式下探针从远离到间歇性接触样品表面,是一个连续的能量耗散过程.针对该连续过程的能量耗散机理研究仅零星存在于各个文献之中,对于连续过程中各个阶段的能量耗散机理也没有一个系统的解释和实验验证.本文提出了新的位移激励下原子力显微镜探针-样品系统简化模型并得到了一维振子系统等效阻尼的计算方法,并通过该方法计算了探针在远离样品表面时的空气黏性阻尼和靠近样品时的空气压膜阻尼,分析了探针从远离样品到间歇性接触样品表面这一过程中的环境耗散机理变化,得到了原子力显微镜系统理论品质因数与探针工作位置的关系曲线;在此基础上设计了轻敲模式下的微悬臂梁扫频实验,得到了系统实验品质因数与探针工作位置的关系曲线,进而验证了理论模型的准确性. 本文通过对轻敲模式下AFM环境耗散机理进行理论分析和实验验证,希望可以对轻敲模式下AFM动力学特性及其阻尼作用机理有更近一步的认识,同时对微纳米机电系统 (MEMS/NEMS) 能量耗散机理的研究提供理论参考和实验方法.     

Complex Dynamics of a Pyranose Ring Structure Molecule Attached to an Atomic Force Microscope

Dynamic numerical simulations were performed for a pyranose ring structure molecule attached to an Atomic Force Microscope (AFM) using a standard semiempirical potential energy surface model. The fundamental static force-extension behavior was first determined using a slow pulling base excitation at the AFM probe. The static force-extension curve displays a stiffness nonlinearity, both softening and hardening, that depends upon level of the pulling force. For the dynamic analysis, a single harmonic base excitation is applied to the AFM probe. A typical evolution process from periodic to aperiodic or chaotic motion obtained by varying the excitation frequency and amplitude is discussed. A strong chaotic response motion was generated for certain system parameters. The numerical analysis shows this chaotic response arises from a molecular structure conformational change.     

SI-Traceable Spring Constant Calibration of Microfabricated Cantilevers for Small Force Measurement

A variety of methods exist to measure the stiffness of microfabricated cantilever beams such as those used as mechanical sensors in atomic force microscopy (AFM). In order for AFM to be used as a quantitative small force measurement tool, these methods must be validated within the International System of Units (SI). To this end, two different contact techniques were used to calibrate the spring constant of a cantilever beam. First, a dynamic indentation-based method was used to measure the spring constant of a rectangular cantilever beam. These results were then compared against an SI-traceable spring constant measurement from an electrostatic force balance (EFB). The measurements agree within experimental uncertainty and within 2% for spring constants greater than 2 N/m. The use of this cantilever beam as a transfer artifact for in situ AFM cantilever calibration was then evaluated in comparison to the thermal calibration method. Excellent agreement is seen between these techniques, establishing the consistency of the thermal and dynamic indentation methods with SI-traceable contact cantilever calibration for the rectangular cantilever geometry tested. Disclaimer: This article is authored by employees of the U.S. federal government, and is not subject to copyright. Commercial equipment and materials are identified in order to adequately specify certain procedures. In no case does such identification imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the materials or equipment identified are necessarily the best available for the purpose.     

In-plane modal testing of a free isotropic rectangular plate

In-plane vibration modes of an aluminum panel were experimentally identified from frequency response tests. Responses were measured on the panel edges and at selected locations on the panel surface. The measurements on the surface were made by attaching accelerometers oriented parallel to the panel plane. Resonance frequencies, relative damping ratios and mode shapes were established for the lowest 12 in-plane modes found in the frequency range between 1600 and 7000 Hz. A damping ratio of less than 0.05 percent of critical damping is proved to be valid for the aluminum panel. A finite element software was used to calculate 12 corresponding theoretical in-plane eigenfrequencies and mode shapes. An outline for a nondestructive procedure is suggested to estimate the input data for the elastic constants of an isotropic plate model. Two of the modes were used in analogy with the flexural vibration of beams and plates. The modes illustrate the deformation pattern including shear deformations, through the thickness, for the bending modes of thick beams or plates. The Rayleigh-Timoshenko theory also was used for the calculation of these two eigenfrequencies.     

原子力显微镜探针振动的简化模型分析

为对原子力显微镜(atomic force microscope,AFM)的微悬臂梁进行定性动力学特性分析,建立AFM微悬臂梁的简化模型,探讨AFM探针的受迫振动.通过理论计算得出AFM探针简化模型的运动方程,并得到振动波形,证明了AFM实际应用中的对称问题和"频漂"问题,并发现AFM简化模型的间歇式碰撞现象.用负弹簧模拟探针针尖与样品之间的长程引力,并通过理论计算探讨长程引力对AFM测量的影响.     

ENERGY DISSIPATION IN TAPPING MODE ATOMIC FORCE MICROSCOPY

原子力显微镜有多种成像模式,其中轻敲模式是最为常用的扫描方式.轻敲模式能获取样品表面形貌的高度信息和相位信息,其中相位信息具有更多的价值,如能反映样品的表面能、弹性、亲疏水性等.依据振动力学理论,相位与振动系统的能量耗散有关.探针样品间的能量耗散对于理解轻敲模式下原子力显微镜的成像机理至关重要,样品特性和测量环境会影响能量耗散.本文在不考虑毛细力影响下,基于JKR接触模型,给出了探针样品相互作用下的加卸载曲线,结合原子力显微镜力曲线实验,给出了探针-样品分离失稳点的位置,从而计算一个完整接触分离过程的能量耗散,进而讨论考虑表面粗糙度对能量耗散的影响.在轻敲模式下考虑毛细力影响,通过特征时间对比,证明挤出效应是液桥生成的主导因素,在等容条件下,用数值方法计算了不同相对湿度对能量耗散的影响.通过一维振子模型,简要说明原子力显微镜相位像与样品表面能、杨氏模量、表面粗糙度、相对湿度之间的关系.分析表明,表面粗糙度和环境湿度均会引起相位的变化,进而认为它们是引起赝像的因素.     

Energy method solution for the postbuckling response of an axially loaded bilaterally constrained beam

Bistable and multistable structures have shown great usefulness in many applications such as MEMS actuation and energy harvesting. Bistability of structures can be achieved through buckling. Confining a buckled beam between two lateral constraints allows it to buckle into higher modes as the axial load increases. This paper presents a theoretical study of the postbuckling response of a bilaterally constrained elastica subjected to gradually increased axial load. Equilibrium states are determined using an energy method. Under small deformation assumptions, the total potential energy is minimized under the defined constraints. The presented model allows for an accurate representation of the flatting behavior and the increase in the length of contact areas with the lateral constraints before the sudden snapping between equilibrium states. Mode transitions are manifested by jumps in the response curves. Previously developed models based on geometry and symmetries overestimate the required forces for higher equilibrium modes and do not match experimental observations. Results are validated with experimental force–displacement measurements under both force- and displacement-control. The kinetic energy released during buckling mode transitions is determined by a dynamic analysis.     

基于修正的Craig-Bampton法的大长径比助推器分离特性研究

在芯级周围布置助推器是提升运载能力的有效手段。对于长径比较高的助推器，结构呈现明显的局部弹性效应，这将对助推器分离的安全性产生影响。针对传统Craig-Bampton方法无法适用于大范围运动弹性体的不足，本文采用修正的Craig-Bampton法获得系统的正则化模态，使研究对象的所有模态与频率一一对应，实现分离体大范围刚体运动与弹性变形的有效耦合。基于该方法开展柔性多体动力学仿真，获得了不同模态阶数和结构刚度的弹性助推器分离系统的分离特性。研究结果表明，与传统的刚体模型相比，考虑助推器弹性效应的分离动力学模型更加贴近真实飞行情况。基于此，获得了各阶模态和结构刚度对分离安全边界的影响规律，其中以一阶横向模态最为敏感。通过研究弹性助推器的分离特性，为运载火箭助推器分离系统的工程实际应用提供理论参考和研究支撑。     

Analytical solutions of the frequency shifts of several modes in AFM scanning an inclined surface,subjected to the Lennard-Jones force

The role of higher cantilever modes is important to obtain some material contrast. The analysis of AFM subjected to a short-range force can improve greatly the studies of surface topography and interaction energies and interaction forces, especially for chemistry and biology materials. When the tip-sample distance is in the order of inter-atomic spacing, the short-range tip-sample force is usually simulated by the Lennard-Jones model. In typical AFM experiments the cantilever is inclined at an angle to the substrate surface. In this study, the analytical method to determine the frequency shift of AFM scanning the relative inclined surface, subjected to the Lennard-Jones force is proposed. The inclined angle may be the angle between the probe and the scanning table or due to the inclined plane and the roughness of sample surface. The closed-form solution of the partial differential equation with a nonlinear boundary condition is derived and then the corresponding frequency shifts of higher modes can be determined easily. The perturbation method transforming a continuous system into the lumped-masses one for determining the frequency shift will lead to an error. Although the above disadvantage exists, the lumped-masses model is simple and intuitive. Therefore, using the principle of dynamic strain energy, the conventional perturbation method is revised to determine the frequency shifts of higher modes in scanning an relative inclined surface. Further, the ratio relation among the frequency shifts is discovered. Moreover, it is found that the effect of the relative inclined angle on the first three frequency shifts is significant.     

Continuum modeling of cell membranes

In this paper, we develop a finite-deformation model for cell membranes with a view toward characterizing the local mechanical response of membranes in atomic force microscope (AFM) experiments. The membrane is modeled as a 2-D fluid continuum endowed with bending resistance. The general theory is used to obtain equations that describe axisymmetric equilibrium states. The membrane is assumed to enclose a fluid medium, which transmits hydrostatic pressure to the membrane, and a point load is applied at the pole to simulate an AFM probe. Both types of loading are associated with a potential and the problem is then cast in a variational setting. The equilibrium equations and boundary conditions are obtained by applying standard variational procedures, resulting in a pair of coupled fourth-order differential equations to be solved for the shape of the meridian. Further refinements associated with global constraints on the enclosed volume and contact with a rigid substrate are introduced, and a solution strategy is proposed which relies on an iterative scheme for calculating the associated Lagrange multipliers.     

Noise influenced elastic cantilever dynamics with nonlinear tip interaction forces

In this work, the authors study the influence of noise on the dynamics of base-excited elastic cantilever structures at the macroscale and microscale by using experimental, numerical, and analytical means. The macroscale system is a base excited cantilever structure whose tip experiences nonlinear interaction forces. These interaction forces are constructed to be similar in form to tip interaction forces in tapping mode atomic force microscopy (AFM). The macroscale system is used to study nonlinear phenomena and apply the associated findings to the chosen AFM application. In the macroscale experiments, the tip of the cantilever structure experiences long-range attractive and short-range repulsive forces. There is a small magnet attached to the tip, and this magnet is attracted by another one mounted to a high-resolution translatory stage. The magnet fixed to the stage is covered by a compliant material that is periodically impacted by the cantilever’s tip. Building on their earlier work, wherein the authors showed that period-doubling bifurcations associated with near-grazing impacts occur during off-resonance base excitations of macroscale and microscale cantilevers, in the present work, the authors focus on studying the influence of Gaussian white noise when it is included as an addition to a deterministic base excitation input. The repulsive forces are modeled as Derjaguin–Muller–Toporov (DMT) contact forces in both the macroscale and microscale systems, and the attractive forces are modeled as van der Waals attractive forces in the microscale system and magnetic attractive forces in the macroscale system. A reduced-order model, based on a single mode approximation is used to numerically study the response for a combined deterministic and random base excitation. It is experimentally and numerically found that the addition of white Gaussian noise to a harmonic base excitation facilitates contact between the tip and the sample, when there was previously no contact with only the harmonic input, and results in a response that is nominally close to a period-doubled orbit. The qualitative change observed with the addition of noise is associated with near-grazing impacts between the tip and the sample. The numerical and experimental results further motivate the formulation of a general analytical framework, in which the Fokker–Planck equation is derived for the cantilever-impactor system. After making a set of approximations, the moment evolution equations are derived from the Fokker–Planck equation and numerically solved. The resulting findings support the experimental results and demonstrate that noise can be added to the input to facilitate contact between the cantilever’s tip and the surface, when there was previously no contact with only a harmonic input. The effects of Gaussian white noise are numerically studied for a tapping mode AFM application, and it is shown that contact between the tip and the sample can be realized by adding noise of an appropriate level to a harmonic excitation.     

多模态原子力显微术研究进展

材料纳米尺度的各种性能中，纳米力学性能是纳米材料和器件服役所需要保证的最基本性能。因此，发展可靠的定量化纳米力学测试技术就显得尤为关键。原子力显微镜(Atomic Force Microscope，AFM)作为纳米力学测试的重要平台，目前广泛应用于材料纳米尺度形貌和力学性能成像。作为原子力显微术的前沿应用模式之一，多模态原子力显微术通过同时激励探针的两个或多个振动模态对样品进行测试或成像，可实现对被测样品高分辨率、高灵敏度、定量化和无损的纳米力学快速成像及检测，具有极其广泛的应用前景。围绕多模态原子力显微术，首先介绍了多模态原子力显微术的基本成像原理和力学模型基础。随后，综述了多模态原子力显微术探针动力学以及成像技术相关研究的主要进展。然后，对多模态原子力显微术的几类典型应用进行了总结和评述。最后，对多模态原子力显微术未来可能的研究方向进行了展望。     

Dynamic operation modes of AFM: Non-linear behavior and theoretical analysis of the stability of the AFM oscillator

The first part of this paper gives a theoretical study of the mechanics of contact of an AFM tip on viscous materials. Analytical expressions are derived showing the non-linear behaviors specifically related to the use of dynamic operation modes of AFM on viscous materials. A detailed analysis of the dissipated energy as a function of the tip indentation is presented. The second part is dedicated to a theoretical analysis investigating the domain of stability of the oscillator and the influence of the machine. The theoretical approach includes the electronic feedback loop used with the frequency modulation mode. Because the interaction between the tip and the sample produces a dynamical non-linear behavior, an unstable branch occurs that can change the stability of the oscillator. In particular, a sudden jump of the oscillating tip can be produced. In spite of the complexity of the problem, the analytical approach ends with two simple equations. The two equations provide an unambiguous way of discriminating between the contributions from the machine and the tip sample interaction.     

Quantifying the nonlinear mode competition in the flow over an open cavity at medium Reynolds number

Our purpose is to quantify the rate of intermittency of nonlinearly competing modes, in a dominantly mode-switching scenario. What is the rate of presence of each mode? Can they simultaneously appear in, or disappear from the signal? The study is done in the context of open flows, exhibiting self-sustained oscillations, where air is here flowing over an open cavity. Reynolds numbers are of the order of 14,000. Velocity measurements downstream of the cavity are based on a laser Doppler velocimetry technique. We propose two methods to estimate the rate of presence of each mode: one based on a complex demodulation technique, the other on the distribution of the state vectors in the phase portrait of the signal.     

Mechanism by which a frictionally confined rod loses stability under initial velocity and position perturbations

We present a theory to reveal for the first time the distinct mechanisms by which a compressed rod confined in a channel buckles in the presence of dry friction. Contrary to the case of a frictionless contact, with friction the system can bear substantially enhanced compressive load without buckling after its stiffness turns negative, and the onset of instability is strongly affected by the amount of perturbation set by the environment. Our theory, confirmed by simulations, shows that friction enhances stability by opening a wide stable zone in the perturbation space. Buckling is initiated when the applied compressive force is such that the boundary of the stable zone touches a point set by the environment, at a much higher critical load. Furthermore, our analysis shows that friction has a strong effect on the buckling mode; an increase in friction is found to lead to higher buckling modes.     

小波分析在悬臂梁裂纹识别中的应用

基于空间信号的小波分析理论,将含裂纹悬臂梁前四阶振型信息直接用于小波变换,小波系数在空间域上的突变反映了裂纹的存在并指出了裂纹的位置.本文分析了前四阶振型对小波识别结果的敏感性,利用小波系数模极大值在尺度上的表现与Lipschitz指数之闻的关系建立了集中因子和裂纹深度之间的关系,以此来估计裂纹深度.鉴于实测信号往往是含噪声信号,分析了噪声对识别结果的影响规律.数值算例表明利用sym4小波对含裂纹梁的四阶振型信息进行小波分析可以准确地识别出裂纹的位置和深度;高阶振型对结构损伤较为敏感,高阶振型更适合于微裂纹和含噪声信息的处理,但高阶振型的非线性也会给裂纹识别带来一定的困难.使用本文方法进行结构裂纹参数识别,噪声对裂纹位置的影响只是指示清晰度的影响,基本不会产生错误的识别,而对裂纹深度的影响远比对位置的影响复杂,由于小波系数混入了噪声成分,从而增加了集中因子的取值,致使识别结果总是比真实结果偏大.     

双介质耦合刚性基弹性层平面应变型导波模式及界面散射能量分配

过渡段动力稳定性问题已成为制约400 km/h及以上高铁路基设计的关键难题,亟需从波动和能量的角度探究由基础非均匀引发的线路系统动力响应放大机理.文章将轨下基础简化为上表面自由、底端固定的刚性基弹性层,将高铁过渡段车致弹性波传播问题提炼为非均匀介质刚性基弹性层中波的散射问题,建立双介质耦合刚性基弹性层平面应变模型,优化该类波导结构频散方程在复平面求根方法,并结合岩土类介质特征展开刚性基弹性层频散分析,以明确其多模式导波特性及散射能量分配,最后,围绕弹性层厚度、刚度比等影响因素开展对比分析.结果表明:刚性基弹性层各模式导波均具有截止频率,弹性层厚度越小,杨氏模量越大,各阶导波模式的截止频率越高;入射波在双介质刚性基弹性层发生散射后,透射场基阶模式导波会占据主体能量,随着高阶导波模式被逐一激发,反射场及透射场高阶模式能量占比会在全频率范围呈现“此消彼长”状态;交换两侧弹性层材料,改变弹性层厚度及两弹性层刚度比不会显著改变能量分布规律,但总体来看,能量更易集中在较软侧弹性层中,各模式导波在激发初始频段会更为活跃,可分配到更多能量.