Dynamic spring constants for higher flexural modes of cantilever plates with applications to atomic force microscopy

In atomic force microscopy (AFM) a sharp tip fixed close to the free end of a cantilever beam interacts with a surface. The interaction can be described by a point-mass model of an equivalent oscillator with a single spring located at the position of the tip. However, other spring constants have to be used to describe the oscillation behavior correctly if forces are acting on the cantilever over an extended lateral range. A point-mass model is then no longer valid. In the present study we derive expressions for the spring constants of cantilevers that can interact with any part of their plan view area along the beam and for all flexural modes. The equations describe the oscillation behavior in the corresponding mass model and are based on the eigenfrequencies and modal shapes of the free cantilever. The results are of high practical relevance, for example if an AFM is operated in a higher flexural mode, if the tip is not located at the free end of the cantilever beam, or if the external conservative forces affecting cantilever movement are not restricted to a single point. The limitations of the approach are discussed.     

Influence of Poisson's ratio variation on lateral spring constant of atomic force microscopy cantilevers

Atomic force microscopy (AFM) can be used to measure the surface morphologies and the mechanical properties of nanostructures. The force acting on the AFM cantilever can be obtained by multiplying the spring constant of AFM cantilever and the corresponding deformation. To improve the accuracy of force experiments, the spring constant of AFM cantilever must be calibrated carefully. Many methods, such as theoretical equations, the finite element method, and the use of reference cantilever, were reported to obtain the spring constant of AFM cantilevers. For the cantilever made of single crystal, the Poisson's ratio varies with different cantilever-crystal angles. In this paper, the influences of Poisson's ratio variation on the lateral spring constant and axial spring constant of rectangular and V-shaped AFM cantilevers, with different tilt angles and normal forces, were investigated by the finite element analysis. When the cantilever's tilt angle is 20 degrees and the Poisson's ratio varies from 0.02 to 0.4, the finite element results show that the lateral spring constants decrease 11.75% for the rectangular cantilever with 1muN landing force and decrease 18.60% for the V-shaped cantilever with 50nN landing force, respectively. The influence of Poisson's ratio variation on axial spring constant is less than 3% for both rectangular and V-shaped cantilevers. As the tilt angle increases, the axial spring constants for rectangular and V-shaped cantilevers decrease substantially. The results obtained can be used to improve the accuracy of the lateral force measurement when using atomic force microscopy.     

Interlaboratory round robin on cantilever calibration for AFM force spectroscopy

Single-molecule force spectroscopy studies performed by Atomic Force Microscopes (AFMs) strongly rely on accurately determined cantilever spring constants. Hence, to calibrate cantilevers, a reliable calibration protocol is essential. Although the thermal noise method and the direct Sader method are frequently used for cantilever calibration, there is no consensus on the optimal calibration of soft and V-shaped cantilevers, especially those used in force spectroscopy. Therefore, in this study we aimed at establishing a commonly accepted approach to accurately calibrate compliant and V-shaped cantilevers. In a round robin experiment involving eight different laboratories we compared the thermal noise and the Sader method on ten commercial and custom-built AFMs. We found that spring constants of both rectangular and V-shaped cantilevers can accurately be determined with both methods, although the Sader method proved to be superior. Furthermore, we observed that simultaneous application of both methods on an AFM proved an accurate consistency check of the instrument and thus provides optimal and highly reproducible calibration. To illustrate the importance of optimal calibration, we show that for biological force spectroscopy studies, an erroneously calibrated cantilever can significantly affect the derived (bio)physical parameters. Taken together, our findings demonstrated that with the pre-established protocol described reliable spring constants can be obtained for different types of cantilevers.     

On a deformation sign for identifying defects on the basis of the analysis of the forms of the natural vibrations of a cantilever with a notch

A method for determining the location and depth of a notch has been developed on the basis of the results of a finite-element analysis of natural vibration modes of a cantilever with a notch. A phenomenological relationship is obtained that relates the degree of cantilever damage to the characteristic features of its natural vibration modes. A new diagnostic sign of the degree of cantilever damage is proposed, which is the angle between the tangents on the plot of the natural form of the fourth vibration mode. An approximate equation is deduced that describes the dependence of the angle between the tangents at the “break” point of the shape of the fourth vibration mode on the depth of a notch in the cantilever.     

Mass determination and sensitivity based on resonance frequency changes of the higher flexural modes of cantilever sensors

Micro- and nanocantilevers are increasingly employed as mass sensors. Most studies consider the first flexural mode and adsorbed masses that are either discretely attached or homogeneously distributed along the entire length of the cantilever. We derive general expressions that allow for the determination of the total attached mass with any mass distribution along the cantilever length and all flexural modes. The expressions are valid for all cantilevers whose flexural deflection can be described by a one-dimensional function. This approach includes the most common types of microcantilevers, namely, rectangular, picket, and V-shaped. The theoretical results are compared with experimental data up to the fourth flexural mode obtained from thermal noise spectra of rectangular and V-shaped cantilevers.     

Atomic force microscopy: loading position dependence of cantilever spring constants and detector sensitivity

A simple and accurate experimental method is described for determining the effective cantilever spring constant and the detector sensitivity of atomic force microscopy cantilevers on which a colloidal particle is attached. By attaching large (approximately 85 microm diameter) latex particles at various positions along the V-shaped cantilevers, we demonstrate how the normal and lateral spring constants as well as the sensitivity vary with loading position. Comparison with an explicit point-load theoretical model has also been used to verify the accuracy of the method.     

基于槽式悬臂梁结构的微质量传感器设计

压电式微质量传感器的测试精度直接依赖于结构频率对质量变化的灵敏程度。本文利用对称槽型梁和压电薄膜组成的对称敏感结构,提出了一种提高传感器灵敏度的结构设计方法,并设计了一种高精度谐振式微质量传感器。建立了结构频率变化对吸附质量敏感性的分析模型,并研究了槽型截面参数、自振频率及振动模态对灵敏度的影响。与矩形截面结构进行了仿真与实验对比,结果表明,相同几何尺寸参数下,槽型截面悬臂梁的一阶自振频率为1 851 Hz,矩形截面悬臂梁的一阶自振频率为1 610 Hz,相应的传感器灵敏度则分别为3.12×10⁴ Hz/g和1.5×10⁴ Hz/g,前者是后者的2倍。该项设计为提高微质量传感器灵敏度提供了一种新思路。     

FORCE TRANSDUCER MODELING OF RECTANGULAR,V–SHAPED,AND DAGGER CANTILEVER PROBES BASED ON ATOMIC FORCE MICROSCOPY

Real-time monitoring of the nano-particle manipulation process by atomic force microscope (AFM) is almost impossible since the manipulator of the AFM is used as either the imaging or manipulation tool at a given instant. As one approach to this problem, researchers scan the area where the target particle exists, before and after the nano-manipulation. Thus, by using some fixed reference features, the new relative position of the particle is obtained from the images. However, this imaging is offline, and unexpected problems during nano-manipulation process cannot be detected. In this article, force transducer of the AFM rectangular, V-shaped, and dagger cantilevers, which convert the three-dimensional deflections measured by the detection system to corresponding three-dimensional force, are modeled and compared. So there would be a feedback from the real-time force system during nano-manipulation process, which can be utilized for better understanding and reliable handling of nano-particles. Furthermore, the manipulation forces are affected directly by spring constants of cantilever. Hence, the cantilever is the most significant and sensitive component of the AFM. Therefore, variations of the lateral, longitudinal, and normal spring constants of the mentioned cantilever for different geometrical parameters are analyzed and discussed.     

Study of the tip mass and interaction force effects on the frequency response and mode shapes of the AFM cantilever

The vibrational characteristics of an atomic force microscope (AFM) cantilever beam play a key role in dynamic mode of the atomic force microscope. As the oscillating AFM cantilever tip approaches the sample, the tip–sample interaction force influences the cantilever dynamics. In this paper, we present a detailed theoretical analysis of the frequency response and mode shape behavior of a cantilever beam in the dynamic mode subject to changes in the tip mass and the interaction regime between the AFM cantilever system and the sample. We consider a distributed parameter model for AFM and use Euler–Bernoulli method to derive an expression for AFM characteristics equation contains tip mass and interaction force terms. We study the frequency response of AFM cantilever under variations of interaction force between AFM tip and sample. Also, we investigate the effect of tip mass on the frequency response and also the quality factor and spring constant of each eigenmodes of AFM micro-cantilever. In addition, the mode shape analysis of AFM cantilever under variations of tip mass and interaction force is investigated. This will incorporate the presentation of explicit analytical expressions and numerical analysis. The results show that by considering the tip mass, the resonance frequencies of the cantilever are decreased. Also, the tip mass has a significant effect on the mode shape of the higher eigenmodes of the AFM cantilever. Moreover, tip mass affects the quality factor and spring constant of each modes.     

On the rational dynamic modes of vibrating machines with an unbalanced vibration exciter of limited power

The aspects of choosing the design parameters of a vibrating machine made according to the scheme of a series two-mass vibrating system, as well as of the modes of its oscillations excited by two synchronously rotating unbalanced vibration exciters with a drive of limited power, are considered. The limitations on possible modes of oscillation excitation caused by taking the designed features of the vibration exciter into account are determined. At the specified masses and stiffnesses of a vibrating system, as well as for the specified parameters of oscillations of the actuating element, the rational values of the parameters of vibration exciter and the damping in the components of the vibrating system and dynamic modes, which are characterized by low expenditures of energy and a small dynamic effort transmitted to the foundation of the machine, are determined. The oscillation modes in which changing the mass of the actuating element leads to the smallest changes of the amplitudes of its oscillations are determined.     

基于固有频率的呼吸式裂纹梁损伤识别方法

将呼吸裂纹梁简化为由扭转弹簧连接的两段弹性梁,在假定振动响应随振幅变化的基础上推导出呼吸裂纹梁的固有频率方程;考虑振动过程中呼吸裂纹的开合情况,假定裂纹梁的刚度是振幅的非线性函数,建立了呼吸裂纹梁的多项式刚度模型;结合等高线裂纹识别理论和方法,提出了一种基于固有频率的呼吸裂纹梁损伤识别方法,算例验证了方法的可行性与有效性。研究表明,该方法的识别精度取决于实验固有频率的精度。     

Modeling and simulation of AFM cantilever with two piezoelectric layers submerged in liquid over rough surfaces

In this paper, the vibration of an atomic force microscope (AFM) cantilever in tapping mode with two whole piezoelectric layers submerged in liquid medium is investigated. In the performed modeling, the sample surface has been considered as rough, and to show these surface asperities, two models of Rumpf and Rabinovich have been employed for analyzing the attractive Van der Waals force. This paper has been organized in two sections. The first section deals with the functioning of cantilever over rough surfaces, which accompanies the changes of the attractive Van der Waals force, and the second section involves the changes in the Van der Waals force which lead to a change in the liquid medium. The cantilever is totally submerged in the liquid. To show the effect of liquid on cantilever, first, only the cantilever tip is immersed in the liquid and it is dynamically analyzed. Then, the cantilever is totally submerged and then taken out of the liquid, so that the additional mass and damping of the cantilever could be calculated. In these two manners of cantilever immersion in liquid, the effects of the added mass and damping on the cantilever can be measured. When a cantilever vibrates totally in liquid, since the mass and damping of the liquid that is present on the cantilever cannot be determined, first, the cantilever's natural frequency in liquid is estimated in the laboratory and then by using this frequency and the cantilever stiffness (which is not medium-dependent and is always considered as constant), the additional mass and damping of the cantilever are determined.     

Coupled lateral bending-torsional vibration sensitivity of atomic force microscope cantilever

We study the influence of the contact stiffness and the ration between cantilever and tip lengths on the resonance frequencies and sensitivities of lateral cantilever modes. We derive expressions to determine both the effective resonance frequency and the mode sensitivity of an atomic force microscope (AFM) rectangular cantilever. Once the contact stiffness is given, the resonance frequency and the sensitivity of the vibration modes can be obtained from the expression. The results show that each mode has a different resonant frequency to variations in contact stiffness and each frequency increased until it eventually reached a constant value at very high contact stiffness. The low-order vibration modes are more sensitive to vibration than the high-order mode when the contact stiffness is low. However, the situation is reversed when the lateral contact stiffness became higher. Furthermore, increasing the ratio of tip length to cantilever length increases the vibration frequency and the sensitivity of AFM cantilever.     

裂纹悬臂梁的扭转弹簧模型及其实验验证

将含裂纹悬臂梁转化为由扭转弹簧联接两段弹性梁构成的连接体,得到理论计算含裂纹梁振动频率的特征方程。确立了求解裂纹梁固有频率的数值计算流程．计算得到了裂纹深度和位置变化时裂纹悬臂梁振动固有频率的变化规律。进行了裂纹悬臂梁的弯曲振动台架实验,验证了本文提出的扭转弹簧模型及固有频率数值计算方法的有效性。     

A new model for investigating the flexural vibration of an atomic force microscope cantilever

A new model for the flexural vibration of an atomic force microscope cantilever is proposed, and a closed-form expression is derived. The effects of angle, damping and tip moment of inertia on the resonant frequency were analysed. Because the tip is not exactly located at one end of the cantilever, the cantilever is modelled as two beams. The results show that the frequency first increases with increase in angle and then decreases to a constant value for high values of the angle. Moreover, the damping is increased at lower contact positions. The tip moment of inertia is also sensitive to the resonant frequency at small values for the odd modes and large values for the even modes.     

不同质量块布局双端固支梁的等效弹簧系数的热效应

通过理论研究和有限元仿真分析了热效应对两种不同质量块布局的双端固支梁的等效弹簧系数的影响,两种质量块分别布局为质量块与梁对称布置和质量块位于梁平面一侧布置。 推导了双端固支梁等效弹簧系数随温度的变化公式,仿真分析了双端固支梁-质量块系统的一阶固有频率以及质量块在静载荷作用下的位移。通过固有频率及载荷-位移两种方式,分别得到了两种布局下的质量系统在不同温度下的等效弹簧系数。对于质量块对称布置和单侧布置,根据一阶固有频率得到的等效弹簧系数随温度的变化与理论分析都吻合;而利用载荷-位移得到的等效弹簧系数随温度的变化,需要对位移进行修正。计算结果表明,对于质量块位于梁一侧的系统,温度变化对其等效弹簧系数有重要影响,质量块的不对称会在温度变化时产生对梁的弯矩作用,从而使质量块产生位移,造成微机电传感器和执行器输出信号的改变。     

基于构型优化的高阶模态微质量传感器灵敏度提升方法

高灵敏度是微质量传感器准确探测细菌、病毒和气体等物质的关键指标。虽然借助微型化的高阶模态梁振动可以有效提升探测灵敏度,但微尺度效应也降低了传感器的抗环境干扰能力。因此,如何在特定尺度约束下提升高阶模态传感器的灵敏度已成为谐振式微传感器设计的前沿问题。本文在研究弹性梁几何构型、压电层尺寸与有效质量分布对振动模态影响关系的基础上,建立了压电驱动多阶梯梁式微质量传感器的灵敏度分析模型,以传感器灵敏度提升最大为目标,建立了高阶振动模态下悬臂梁几何构型优化设计模型,得到了在不同振动模态下具有最高灵敏度的悬臂梁构型,使同尺寸传感器的灵敏度提升了10.0~15.0倍。考虑驱动位置与制造成本约束,设计并研制了具有六阶梯梁结构的高阶模态微质量传感器。实验结果表明,总长度为17.6mm的六阶梯梁微质量传感器的灵敏度为18.8×10~4 Hz/g,考虑制造误差的影响,其二阶模态灵敏度为同尺寸等截面梁传感器的10.0倍,较一阶模态同尺寸传感器灵敏度提升了19.8倍,从而验证了所提出的高阶模态微质量传感器灵敏度提升方法的有效性和可行性。     

Cantilever spring constant calibration using laser Doppler vibrometry

Uncertainty in cantilever spring constants is a critical issue in atomic force microscopy (AFM) force measurements. Though numerous methods exist for calibrating cantilever spring constants, the accuracy of these methods can be limited by both the physical models themselves as well as uncertainties in their experimental implementation. Here we report the results from two of the most common calibration methods, the thermal tune method and the Sader method. These were implemented on a standard AFM system as well as using laser Doppler vibrometry (LDV). Using LDV eliminates some uncertainties associated with optical lever detection on an AFM. It also offers considerably higher signal to noise deflection measurements. We find that AFM and LDV result in similar uncertainty in the calibrated spring constants, about 5%, using either the thermal tune or Sader methods provided that certain limitations of the methods and instrumentation are observed.     

星形人字齿轮传动系统振动模式的理论研究

采用集中质量法建立了含N个星轮的星形人字齿轮传动系统的弯-扭-轴耦合动力学模型和动力学方程。模型中将人字齿轮看作由两斜齿轮通过无质量弹簧连接而成。求解了系统的固有圆频率和固有振型,根据振动模式将星形人字齿轮传动系统的固有振动模式进行分类,并对其重根数和固有频数个数进行了理论研究。结果表明：星形人字齿轮传动系统含有4种典型振动模式,即：中心构件轴向-扭转耦合模态(重根数 r=1),星轮模态(重根数 r=N-3,N>3),中心构件平动模态(重根数 r=2),星轮与中心轮耦合模态(重根数 r=2)。     

Spring constant calibration of atomic force microscopy cantilevers with a piezosensor transfer standard

We describe a method to calibrate the spring constants of cantilevers for atomic force microscopy (AFM). The method makes use of a "piezosensor" composed of a piezoresistive cantilever and accompanying electronics. The piezosensor was calibrated before use with an absolute force standard, the NIST electrostatic force balance (EFB). In this way, the piezosensor acts as a force transfer standard traceable to the International System of Units. Seven single-crystal silicon cantilevers with rectangular geometries and nominal spring constants from 0.2 to 40 Nm were measured with the piezosensor method. The values obtained for the spring constant were compared to measurements by four other techniques: the thermal noise method, the Sader method, force loading by a calibrated nanoindentation load cell, and direct calibration by force loading with the EFB. Results from different methods for the same cantilever were generally in agreement, but differed by up to 300% from nominal values. When used properly, the piezosensor approach provides spring-constant values that are accurate to +/-10% or better. Methods such as this will improve the ability to extract quantitative information from AFM methods.