A new calibration method of the lateral contact stiffness and lateral force using modulated lateral force microscopy

We describe a new calibration method for lateral contact stiffness using modulated lateral force microscopy, a technique that offers some advantages with respect to the more classical friction force microscopy currently used for characterizing the friction properties of materials. The calibration method is based on the study of the lateral contact stiffness versus applied load and on the use of elasticity contact theories to determine by fit the calibration coefficient, allowing the scaling of experimental data. The method is tested by measuring the friction coefficient and shear strength of silicon and mica samples, respectively, and compared with results from the literature. This revised version was published online in August 2006 with corrections to the Cover Date.     

Quantitative comparison of two independent lateral force calibration techniques for the atomic force microscope

Two independent lateral-force calibration methods for the atomic force microscope (AFM)--the hammerhead (HH) technique and the diamagnetic lateral force calibrator (D-LFC)--are systematically compared and found to agree to within 5?% or less, but with precision limited to about 15?%, using four different tee-shaped HH reference probes. The limitations of each method, both of which offer independent yet feasible paths toward traceable accuracy, are discussed and investigated. We find that stiff cantilevers may produce inconsistent D-LFC values through the application of excessively high normal loads. In addition, D-LFC results vary when the method is implemented using different modes of AFM feedback control, constant height and constant force modes, where the latter is more consistent with the HH method and closer to typical experimental conditions. Specifically, for the D-LFC apparatus used here, calibration in constant height mode introduced errors up to 14?%. In constant force mode using a relatively stiff cantilever, we observed an ≈?4?% systematic error per μN of applied load for loads ≤?1 μN. The issue of excessive load typically emerges for cantilevers whose flexural spring constant is large compared with the normal spring constant of the D-LFC setup (such that relatively small cantilever flexural displacements produce relatively large loads). Overall, the HH method carries a larger uncertainty, which is dominated by uncertainty in measurement of the flexural spring constant of the HH cantilever as well as in the effective length dimension of the cantilever probe. The D-LFC method relies on fewer parameters and thus has fewer uncertainties associated with it. We thus show that it is the preferred method of the two, as long as care is taken to perform the calibration in constant force mode with low applied loads.     

Lateral force microscope calibration using a modified atomic force microscope cantilever

A proof-of-concept study is presented for a prototype atomic force microscope (AFM) cantilever and associated calibration procedure that provide a path for quantitative friction measurement using a lateral force microscope (LFM). The calibration procedure is based on the method proposed by Feiler et al. [Rev. Sci. Instrum. 71, 2746 (2000)] but allows for calibration and friction measurements to be carried out in situ and with greater precision. The modified AFM cantilever is equipped with lateral lever arms that facilitate the application of normal and lateral forces, comparable to those acting in a typical LFM friction experiment. The technique allows the user to select acceptable precision via a potentially unlimited number of calibration measurements across the full working range of the LFM photodetector. A microfabricated version of the cantilever would be compatible with typical commercial AFM instrumentation and allow for common AFM techniques such as topography imaging and other surface force measurements to be performed.     

Calibration of lateral force measurements in atomic force microscopy with a piezoresistive force sensor

We present here a method to calibrate the lateral force in the atomic force microscope. This method makes use of an accurately calibrated force sensor composed of a tipless piezoresistive cantilever and corresponding signal amplifying and processing electronics. Two ways of force loading with different loading points were compared by scanning the top and side edges of the piezoresistive cantilever. Conversion factors between the lateral force and photodiode signal using three types of atomic force microscope cantilevers with rectangular geometries (normal spring constants from 0.092 to 1.24 N/m and lateral stiffness from 10.34 to 101.06 N/m) were measured in experiments using the proposed method. When used properly, this method calibrates the conversion factors that are accurate to +/-12.4% or better. This standard has less error than the commonly used method based on the cantilever's beam mechanics. Methods such of this allow accurate and direct conversion between lateral forces and photodiode signals without any knowledge of the cantilevers and the laser measuring system.     

Improved lateral force calibration based on the angle conversion factor in atomic force microscopy

A novel calibration method is proposed for determining lateral forces in atomic force microscopy (AFM), by introducing an angle conversion factor, which is defined as the ratio of the twist angle of a cantilever to the corresponding lateral signal. This factor greatly simplifies the calibration procedures. Once the angle conversion factor is determined in AFM, the lateral force calibration factors of any rectangular cantilever can be obtained by simple computation without further experiments. To determine the angle conversion factor, this study focuses on the determination of the twist angle of a cantilever during lateral force calibration in AFM. Since the twist angle of a cantilever cannot be directly measured in AFM, the angles are obtained by means of the moment balance equations between a rectangular AFM cantilever and a simple commercially available step grating. To eliminate the effect of the adhesive force, the gradients of the lateral signals and the twist angles as a function of normal force are used in calculating the angle conversion factor. To verify reliability and reproducibility of the method, two step gratings with different heights and two different rectangular cantilevers were used in lateral force calibration in AFM. The results showed good agreement, to within 10%. This method was validated by comparing the coefficient of friction of mica so determined with values in the literature.     

Quantitative elastic modulus measurement by magnetic force modulation microscopy

It is experimentally demonstrated that magnetic force modulation microscopy (MFMM) is a technique allowing quantitative elastic modulus measurements. A model of the cantilever–tip–sample interaction taking into account the lateral contact stiffness (i.e., the friction effects at the level of the tip–sample contact), the position of the magnetic force applied to the cantilever with respect to the tip position, as well as the inclination of the cantilever arm with respect to the sample surface is presented. The model shows that MFMM is much less sensitive to lateral force than the other modulation techniques and thus, in contrast to the latter, that the contrast of the stiffness images can be interpreted as a true elasticity contrast and not as a mixture of friction and elasticity. Thanks to the study of the normal contact stiffness versus normal load that allows the characterization of contact between tip and sample, it is possible to perform quantitative elastic modulus measurements with a dynamic modulation method.     

Prototype cantilevers for quantitative lateral force microscopy

Prototype cantilevers are presented that enable quantitative surface force measurements using contact-mode atomic force microscopy (AFM). The "hammerhead" cantilevers facilitate precise optical lever system calibrations for cantilever flexure and torsion, enabling quantifiable adhesion measurements and friction measurements by lateral force microscopy (LFM). Critically, a single hammerhead cantilever of known flexural stiffness and probe length dimension can be used to perform both a system calibration as well as surface force measurements in situ, which greatly increases force measurement precision and accuracy. During LFM calibration mode, a hammerhead cantilever allows an optical lever "torque sensitivity" to be generated for the quantification of LFM friction forces. Precise calibrations were performed on two different AFM instruments, in which torque sensitivity values were specified with sub-percent relative uncertainty. To examine the potential for accurate lateral force measurements using the prototype cantilevers, finite element analysis predicted measurement errors of a few percent or less, which could be reduced via refinement of calibration methodology or cantilever design. The cantilevers are compatible with commercial AFM instrumentation and can be used for other AFM techniques such as contact imaging and dynamic mode measurements.     

Stick–slip motion in the atomic force microscope

Measurements of atomic friction in the atomic force microscope frequently show periodic variations at the lattice spacing of the surface being scanned, which have the saw‐tooth wave form characteristic of “stick–slip” motion. Simple models of this behaviour have been proposed, in which the “dynamic element” of the system is provided by the elastic stiffness and inertia of the cantilever which supports the tip of the microscope, in its lateral, i.e., torsional mode of vibration. These models have been successful in predicting the observed motion, but only by assuming that the cantilever is heavily damped. However, the source of this damping in a highly elastic cantilever is not explained. To resolve the paradox, it is shown in this note that it is necessary to introduce the elastic stiffness of the contact into the model. The relationship between the contact stiffness, the cantilever stiffness and the amplitude of the periodic friction force is derived in order for stick–slip motion at lattice spacing to be achieved. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

Dynamic analysis of tapping mode atomic force microscope (AFM) for critical dimension measurement

Transient dynamics of tapping mode atomic force microscope (AFM) for critical dimension measurement are analyzed. A simplified nonlinear model of AFM is presented to describe the forced vibration of the micro cantilever-tip system with consideration of both contact and non-contact transient behavior for critical dimension measurement. The governing motion equations of the AFM cantilever system are derived from the developed model. Based on the established dynamic model, motion state of the AFM cantilever system is calculated utilizing the method of averaging with the form of slow flow equations. Further analytical solutions are obtained to reveal the effects of critical parameters on the system dynamic performance. In addition, features of dynamic response of tapping mode AFM in critical dimension measurement are studied, where the effects of equivalent contact stiffness, quality factor and resonance frequency of cantilever on the system dynamic behavior are investigated. Contact behavior between the tip and sample is also analyzed and the frequency drift in contact phase is further explored. Influence of the interaction between the tip and sample on the subsequent non-contact phase is studied with regard to different parameters. The dependence of the minimum amplitude of tip displacement and maximum phase difference on the equivalent contact stiffness, quality factor and resonance frequency are investigated. This study brings further insights into the dynamic characteristics of tapping mode AFM for critical dimension measurement, and thus provides guidelines for the high fidelity tapping mode AFM scanning.     

Quantification of the lateral detachment force for bacterial cells using atomic force microscope and centrifugation

To determine the lateral detachment force for individual bacterial cells, a quantitative method using the contact mode of an atomic force microscope (AFM) was developed in this study. Three key factors for the proposed method, i.e. scan size, scan rate and cantilever choice, were evaluated and optimized. The scan size of 40×40 μm² was optimal for capturing sufficient number of adhered cells in a microscopic field and provide adequate information for cell identification and detachment force measurement. The scan rate affected the measurement results significantly, and was optimized at 40 μm/s considering both force measurement accuracy and experimental efficiency. The hardness of applied cantilevers also influenced force determination. The proposed protocol for cantilever selection is to use those with the lowest spring constant first and then step up to a harder cantilever until all cells are detached. The lateral detachment force of Escherichia coli cells on polished stainless steel and a glass-slide coated with poly-l-lysine were measured as 0.763±0.167 and 0.639±0.136 nN, respectively. The results showed that the established method had good repeatability and sensitivity to various bacteria/substrata combinations. The detachment force quantified by AFM (0.639±0.136 nN) was comparable to that measured by the centrifugation method (1.12 nN).     

Lateral force microscopy of multiwalled carbon nanotubes

Carbon nanotubes are usually imaged with the atomic force microscope (AFM) in non-contact mode. However, in many applications, such as mechanical manipulation or elasticity measurements, contact mode is used. The forces affecting the nanotube are then considerable and not fully understood. In this work lateral forces were measured during contact mode imaging with an AFM across a carbon nanotube. We found that, qualitatively, both magnitude and sign of the lateral forces to the AFM tip were independent of scan direction and can be concluded to arise from the tip slipping on the round edges of the nanotube. The dependence on the normal force applied to the tip and on the ratio between nanotube diameter and tip radius was studied. We show that for small values of this ratio, the lateral force signal can be explained with a simple geometrical model.     

用复合振子模型计算纳米尺度滑动摩擦力的研究

以光滑界面摩擦为研究对象,探讨用复合振子模型计算纳米尺度滑动摩擦力的原理和方法,推导出滑动摩擦力和摩擦因数的计算公式,并用原子力显微镜探针在硅试样和云母试样上做扫描实验进行验证。实验值与理论计算值的对比结果表明,两者所反映的规律基本一致,表明所提出的理论和方法可行。研究表明,滑动摩擦力与接触面积成线性增长关系,并随宏观振子横向刚度的增大而减小,由于可通过改变摩擦表面的几何形貌来改变宏观振子的横向刚度,因此这一结论将为摩擦控制提供一种新途径。     

Dual resonance excitation system for the contact mode of atomic force microscopy

We propose an improved system that enables simultaneous excitation and measurements of at least two resonance frequency spectra of a vibrating atomic force microscopy (AFM) cantilever. With the dual resonance excitation system it is not only possible to excite the cantilever vibrations in different frequency ranges but also to control the excitation amplitude for the individual modes. This system can be used to excite the resonance frequencies of a cantilever that is either free of the tip-sample interactions or engaged in contact with the sample surface. The atomic force acoustic microscopy and principally similar methods utilize resonance frequencies of the AFM cantilever vibrating while in contact with the sample surface to determine its local elastic modulus. As such calculation demands values of at least two resonance frequencies, two or three subsequent measurements of the contact resonance spectra are necessary. Our approach shortens the measurement time by a factor of two and limits the influence of the AFM tip wear on the values of the tip-sample contact stiffness. In addition, it allows for in situ observation of processes transpiring within the AFM tip or the sample during non-elastic interaction, such as tip fracture.     

Atomistic study of lateral contact stiffness in friction force microscopy

The effective stiffness of a friction force microscope tip–substrate system is an important parameter that describes the relationship between lateral force and elastic deformation. In this study, we use a multi-spring model to simplify the system, where two contributions, the tip apex stiffness and the lateral contact stiffness, are discussed in detail. Molecular dynamics simulations are used to characterize stiffness by simulating a tip apex subject to shear or sliding over a substrate surface. The results show that, although the height of the tip apex and tip–substrate orientation affect the various stiffness contributions, the contact itself dominates the overall compliance.     

基于落锤装置的力监测压力绝对校准方法研究

在详细探讨了利用落锤装置进行压力校准的多种途径的基础上,针对力监测压力方法中存在的传感器力值测量精度易受预紧力和惯性力影响的问题,提出了一种改进方法,即在落锤装置原有锤头结构的基础上自行研制了高精度的应变式力传感器,推导了通过自研力传感器监测造压油缸内压力的惯性力修正模型,并总结了模型进一步简化所需满足的前提条件。在此基础上分别对传统和自研力传感器开展了多组校准试验,试验结果表明,采用过渡件和螺栓固定等传统安装方式会对力传感器产生附加预紧力和惯性力两方面的不良影响,造成传感器在测量锤头和活塞杆之间的撞击力时产生很大的测试误差;自研力传感器在满足重锤质量远大于活塞杆质量并保证撞击对中精度的前提下,能够直接根据传感器测得的力值和活塞杆横截面积计算出缸内的压力,和现有的比对式准静态校准方法相比,其高压段压力监测精度优于1%,可作为一种有效的压力校准方法。     

Precise atomic force microscope cantilever spring constant calibration using a reference cantilever array

A method for calibrating the stiffness of atomic force microscope (AFM) cantilevers is demonstrated using an array of uniform microfabricated reference cantilevers. A series of force-displacement curves was obtained using a commercial AFM test cantilever on the reference cantilever array, and the data were analyzed using an implied Euler-Bernoulli model to extract the test cantilever spring constant from linear regression fitting. The method offers a factor of 5 improvement over the precision of the usual reference cantilever calibration method and, when combined with the Systeme International traceability potential of the cantilever array, can provide very accurate spring constant calibrations.     

Combined quantitative ultrasonic and time-resolved interaction force AFM imaging

The authors describe a method where quantitative ultrasonic atomic force microscopy (UAFM) is achieved during time-resolved interaction force (TRIF) imaging in intermittent contact mode. The method uses a calibration procedure for quantitative UAFM. It improves elasticity measurements of stiff regions of surfaces while retaining the capabilities of the TRIF mode for topography, adhesion, dissipation, and elasticity measurements on soft regions of sample surfaces. This combination is especially advantageous when measuring and imaging samples with broad stiffness range in a nondestructive manner. The experiments utilize an active AFM probe with high bandwidth and the UAFM calibration is performed by measuring the magnitude of the time-resolved UAFM signal at a judiciously chosen frequency for different contact stiffness values during individual taps. Improved sensitivity to stiff surface elasticity is demonstrated on a special sample. The results show that combining UAFM with TRIF provides 2.5 GPa (5%) standard deviation on the silicon surface reduced Young's modulus, representing 5× improvement over using only TRIF mode imaging.     

压电力传感器晶片与电极接触刚度影响研究*

以压电力传感器为核心的多分量测量系统是航空、航天等领域重要的力学测量装置。压电力传感器中晶片与电极结合面接触刚度是基础性能参数,对压电力传感器的分载效应及灵敏度,多分量测量系统的整体刚度及静、动态性能有直接影响。由于尚无对该参数的研究,导致在设计封装压电力传感器过程中缺少相应的理论依据,在使用多分量测量系统时存在标定困难等问题。针对以上问题,研究晶片与电极结合面接触刚度的影响因素,建立基于分形理论的接触刚度模型,优化了晶面表面形貌、压电材料、电极材料、预紧力等参数。在此基础上,研究接触刚度对传感器性能——刚度、灵敏度、固有频率等影响,提出一种以接触刚度为约束的晶片表面形貌优化方法。基于结构函数法,建立分形理论与实际工程测量参数的联系,通过两种压电晶体——石英与硅酸镓镧,四种电极材料——钛合金、不锈钢、黄铜、铝,试验验证了理论模型。试验表明,优化晶片表面分形参数、提高预紧力、选用弹性模量小的电极有助于提高晶片与电极接触刚度,进而提高传感器的整体刚度、灵敏度和固有频率。研究为高性能压电力传感器的设计提供了理论参考。     

Force modulation with a scanning force microscope: an analysis

A magnetic force modulation microscope (FMM) has been employed to measure the dynamic behavior of a contact between a scanning force microscope (SFM) tip and a surface. Our experimental results show the inefficiency of the classical models (two Kelvin-Voigt elements in parallel). A new model which takes into account the normal and tangential stiffness of the contact, and also the geometrical and mechanical properties of the cantilever which hold the tip, is proposed. This model shows that the natural frequency is sensitive to the normal stiffness, only if the ratio of the normal contact stiffness to the cantilever stiffness is between 0.2 and 200. Above this domain, the natural frequency is sensitive to sliding (Mindlin theory).