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.     

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.     

Development of eddy current microscopy for high resolution electrical conductivity imaging using atomic force microscopy

We present a high resolution electrical conductivity imaging technique based on the principles of eddy current and atomic force microscopy (AFM). An electromagnetic coil is used to generate eddy currents in an electrically conducting material. The eddy currents generated in the conducting sample are detected and measured with a magnetic tip attached to a flexible cantilever of an AFM. The eddy current generation and its interaction with the magnetic tip cantilever are theoretically modeled using monopole approximation. The model is used to estimate the eddy current force between the magnetic tip and the electrically conducting sample. The theoretical model is also used to choose a magnetic tip-cantilever system with appropriate magnetic field and spring constant to facilitate the design of a high resolution electrical conductivity imaging system. The force between the tip and the sample due to eddy currents is measured as a function of the separation distance and compared to the model in a single crystal copper. Images of electrical conductivity variations in a polycrystalline dual phase titanium alloy (Ti-6Al-4V) sample are obtained by scanning the magnetic tip-cantilever held at a standoff distance from the sample surface. The contrast in the image is explained based on the electrical conductivity and eddy current force between the magnetic tip and the sample. The spatial resolution of the eddy current imaging system is determined by imaging carbon nanofibers in a polymer matrix. The advantages, limitations, and applications of the technique are discussed.     

Measurement sensitivity improvement in tapping-mode atomic force microscopy through bi-harmonic drive signal

This article presents a novel method to improve the measurement sensitivity and reduce impact forces in tapping-mode atomic force microscopy by reshaping the tip trajectory. A tapping drive signal composed of two harmonics is used to generate an oscillating trajectory with a broader valley compared to the typical sinusoidal trajectory. The wide broad valley reduces the velocity of the tip in the vicinity of the sample and allots a greater portion of each period in the vicinity of the sample. Numerical simulations show that this results in decreased impact force and increased sensitivity of the cantilever oscillation to changes in tip-sample offset. Experimental results demonstrate an increase in image sharpness and decrease in tip wear using the bi-harmonic driving signal.     

Dynamic response of rotating flexible cantilever pipe conveying fluid with tip mass

In this study the vibration system is consisted of a rotating cantilever pipe conveying fluid and a tip mass. The equation of motion is derived by using the Lagrange's equation. Also, the equation of motion is derived applying a modeling method that employs hybrid deformation variables. The influences of the rotating angular velocity and the velocity of fluid flow on the dynamic behavior of a cantilever pipe are studied by the numerical method. The effects of a tip mass on the dynamic behavior of a rotating cantilever pipe are also studied. The influences of a tip mass, the velocity of fluid, the angular velocity of a cantilever pipe and the coupling of these factors on the dynamic behavior of a cantilever pipe are analytically clarified. The natural frequencies of a cantilever pipe conveying fluid are proportional to the angular velocity of the pipe and a tip mass in both axial direction and lateral direction.     

Effect of cantilevers' dimensions on phase contrast in multifrequency atomic force microscopy

During the past years, different theoretical and experimental works are done to enhance the observables (mostly higher eigenmode's phase contrast) in multifrequency atomic force microscopy methods. In this study, the geometry of rectangular cantilevers is studied and an optimum dimension that can provide maximum phase contrast for a given set of samples is found. The analysis is done both numerically and experimentally. A sensitivity analysis is provided to demonstrate which dimension (length, width, thickness, tip‐radius, and cantilever and sample angle) of the cantilever has a higher effect on the results. The effects of geometrical dimensions are categorized into to: (a) effect on dynamics of the cantilever (b) effects on cantilever's specifications (i.e., spring constant and quality factor). Length and width of the cantilever dominates the static behavior of the cantilever. While thickness (for lower values), tip radius, and approach angle mostly affect the dynamic behavior of the cantilever. Theoretically, it is found as the length increases the phase contrast increase. This relationship is opposite for width. It was also observed that the effect of thickness for a specific range on the phase contrast depends on the 1st eigenmode amplitude setpoint. This study shows for having higher contrast, lower tip‐radius is needed. The optimum angle between cantilever and sample to enhance bimodal atomic force microscopy imaging is also found. Based on the commercially available cantilevers, the optimum cantilever dimension is provided. Three different cantilevers with similar dimensions are experimentally tested and theoretical results are verified.     

Piezoresistive cantilever force-clamp system

We present a microelectromechanical device-based tool, namely, a force-clamp system that sets or "clamps" the scaled force and can apply designed loading profiles (e.g., constant, sinusoidal) of a desired magnitude. The system implements a piezoresistive cantilever as a force sensor and the built-in capacitive sensor of a piezoelectric actuator as a displacement sensor, such that sample indentation depth can be directly calculated from the force and displacement signals. A programmable real-time controller operating at 100 kHz feedback calculates the driving voltage of the actuator. The system has two distinct modes: a force-clamp mode that controls the force applied to a sample and a displacement-clamp mode that controls the moving distance of the actuator. We demonstrate that the system has a large dynamic range (sub-nN up to tens of μN force and nm up to tens of μm displacement) in both air and water, and excellent dynamic response (fast response time, <2 ms and large bandwidth, 1 Hz up to 1 kHz). In addition, the system has been specifically designed to be integrated with other instruments such as a microscope with patch-clamp electronics. We demonstrate the capabilities of the system by using it to calibrate the stiffness and sensitivity of an electrostatic actuator and to measure the mechanics of a living, freely moving Caenorhabditis elegans nematode.     

Analysis of dynamic cantilever behavior in tapping mode atomic force microscopy

Tapping mode atomic force microscopy (AFM) provides phase images in addition to height and amplitude images. Although the behavior of tapping mode AFM has been investigated using mathematical modeling, comprehensive understanding of the behavior of tapping mode AFM still poses a significant challenge to the AFM community, involving issues such as the correct interpretation of the phase images. In this paper, the cantilever's dynamic behavior in tapping mode AFM is studied through a three dimensional finite element method. The cantilever's dynamic displacement responses are firstly obtained via simulation under different tip‐sample separations, and for different tip‐sample interaction forces, such as elastic force, adhesion force, viscosity force, and the van der Waals force, which correspond to the cantilever's action upon various different representative computer‐generated test samples. Simulated results show that the dynamic cantilever displacement response can be divided into three zones: a free vibration zone, a transition zone, and a contact vibration zone. Phase trajectory, phase shift, transition time, pseudo stable amplitude, and frequency changes are then analyzed from the dynamic displacement responses that are obtained. Finally, experiments are carried out on a real AFM system to support the findings of the simulations. Microsc. Res. Tech. 78:935–946, 2015. © 2015 Wiley Periodicals, Inc.     

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.     

Analysis of the effect of mechanical properties of liquid and geometrical parameters of cantilever on the frequency response function of AFM

The dynamic behavior of atomic force microscopy (AFM) cantilevers in liquid is completely different from its behavior in air due to the applied hydrodynamic force. Exciting cantilever with frequencies close to resonance frequency and primary position alignment are two critical issues that should be considered in deriving frequency response function (FRF). In this paper, the hydrodynamic force has been modeled with string of spheres and the effect of the damping and the added mass on the model has been analyzed. Afterward, this force is applied to the dynamic equation so that the dynamic behavior of AFM cantilevers is studied in liquids by analyzing the effect of some important parameters such as added mass, internal, and fluid damping. By simulations of the dynamic equations for a silicon cantilever, FRF is determined in both air and liquid. In addition, the effects of two significant parameters of liquid mechanical properties (liquid viscosity and density) and geometrical parameters of cantilever on FRF are studied. The results for string of spheres model are compared with the other hydrodynamic model and the experimental data. When length/width ratio decreases, it is found that string of spheres model has a better agreement than the other hydrodynamic model with experimental data.     

Computational models of a nano probe tip for static behaviors

It is difficult to predict the measurement bias arising from the compliance of the atomic force microscope (AFM) probe. The issue becomes particularly important in this situation where nanometer uncertainties are sought for measurements with dimensional probes composed of flexible carbon nanotubes mounted on AFM cantilevers. We have developed a finite element model for simulating the mechanical behavior of AFM cantilevers with carbon nanotubes attached. Spring constants of both the nanotube and cantilever in two directions are calculated using the finite element method with known Young's moduli of both silicon and multiwall nanotube as input data. Compliance of the nanotube-attached AFM probe tip may be calculated from the set of spring constants. This paper presents static models that together provide a basis to estimate uncertainties in linewidth measurement using nanotubes. In particular, the interaction between a multiwall nanotube tip and a silicon sample is modeled using the Lennard-Jones theory. Snap-in and snap-out of the probe tip in a scanning mode are calculated by integrating the compliance of the probe and the sample-tip interacting force model. Cantilever and probe tip deflections and points of contact are derived for both horizontal scanning of a plateau and vertically scanning of a wall. The finite element method and the Lennard-Jones model provide a means to analyze the interaction of the probe and sample and measurement uncertainty, including actual deflection and the gap between the probe tip and the measured sample surface.     

Non-optical bimorph-based tapping-mode force sensing method for scanning near-field optical microscopy

A non‐optical bimorph‐based tapping‐mode force sensing method for tip–sample distance control in scanning near‐field optical microscopy is developed. Tapping‐mode force sensing is accomplished by use of a suitable piezoelectric bimorph cantilever, attaching an optical fibre tip to the extremity of the cantilever free end and fixing the guiding portion of the fibre to a stationary part near the tip to decouple it from the cantilever. This method is mainly characterized by the use of a bimorph, which carries out simultaneous excitation and detection of mechanical vibration at its resonance frequency owing to piezoelectric and anti‐piezoelectric effects, resulting in simplicity, compactness, ease of implementation and lack of parasitic optical background. In conjugation with a commercially available SPM controller, tapping‐mode images of various samples, such as gratings, human breast adenocarcinoma cells, red blood cells and a close‐packed layer of 220‐nm polystyrene spheres, have been obtained. Furthermore, topographic and near‐field optical images of a layer of polystyrene spheres have also been taken simultaneously. The results suggest that the tapping‐mode set‐up described here is reliable and sensitive, and shows promise for biological applications.     

A method to provide rapid in situ determination of tip radius in dynamic atomic force microscopy

We provide a method to characterize the tip radius of an atomic force microscopy in situ by monitoring the dynamics of the cantilever in ambient conditions. The key concept is that the value of free amplitude for which transitions from the attractive to repulsive force regimes are observed, strongly depends on the curvature of the tip. In practice, the smaller the value of free amplitude required to observe a transition, the sharper the tip. This general behavior is remarkably independent of the properties of the sample and cantilever characteristics and shows the strong dependence of the transitions on the tip radius. The main advantage of this method is rapid in situ characterization. Rapid in situ characterization enables one to continuously monitor the tip size during experiments. Further, we show how to reproducibly shape the tip from a given initial size to any chosen larger size. This approach combined with the in situ tip size monitoring enables quantitative comparison of materials measurements between samples. These methods are set to allow quantitative data acquisition and make direct data comparison readily available in the community.     

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.     

Improvement of KFM performance by intermittent bias application method and by sampling detection of cantilever deflection

We have proposed an intermittent bias application method as well as a sampling detection method of cantilever deflection in Kelvin probe force microscopy (KFM) to improve its performances for surface potential measurements. In the former method, spiky biases, instead of the bias in a sinusoidal waveform normally used in KFM, are intermittently applied to generate electrostatic force at exact moments when the tip approaches the closest position to a sample surface. The latter one, on the other hand, realizes very sensitive detection of the electrostatic force, which is preferable in KFM. Both the dependence of the electrostatic force on the dc offset bias and the observed potential images clearly indicate that these two methods are very effective to improve the KFM performance.     

Influence of the tip mass on the tip-sample interactions in TM-AFM

This paper focuses on the influences of the tip mass ratio (the ratio of the tip mass to the cantilever mass), on the excitation of higher oscillation eigenmodes and also on the tip-sample interaction forces in tapping mode atomic force microscopy (TM-AFM). A precise model for the cantilever dynamics capable of accurate simulations is essential for the investigation of the tip mass effects on the interaction forces. In the present work, the finite element method (FEM) is used for modeling the AFM cantilever to consider the oscillations of higher eigenmodes oscillations. In addition, molecular dynamics (MD) is used to calculate precise data for the tip-sample force as a function of tip vertical position with respect to the sample. The results demonstrate that in the presence of nonlinear tip-sample interaction forces, the tip mass ratio plays a significant role in the excitations of higher eigenmodes and also in the normal force applied on the surface. Furthermore, it has been shown that the difference between responses of the FEM and point-mass models in different system operational conditions is highly affected by the tip mass ratio.     

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).     

Direct probing of solvent-induced charge degradation in polypropylene electret fibres via electrostatic force microscopy

Electrostatic force microscopy was used to directly probe solvent‐induced charge degradation in electret filter media. Electrostatic force gradient images of individual polypropylene electret fibres were used to quantify the extent of charge degradation caused by the immersion of the fibres into isopropanol. Electrostatic force gradient images were obtained by monitoring the shifts in phase and frequency between the oscillations of the biased atomic force microscopy (AFM) cantilever and those of the piezoelectric driver. Electrostatic force microscopy measurements were performed using non‐contact scans at a constant tip‐sample separation of 75 nm with varied bias voltages applied to the cantilever. Mathematical expressions, based on the capacitance of the tip‐sample system, were used to model the phase and frequency shifts as functions of the applied bias voltage to the tip and the offset voltage due to the fibre's charge. Quantitative agreement between the experimental data and the simplified model was observed.     

Atomic force microscopy imaging of living cells: a preliminary study of the disruptive effect of the cantilever tip on cell morphology

Recent studies have demonstrated that atomic force microscopy (AFM) is a potential tool for studying important dynamic cellular processes in real time. However, the interactions between the cantilever tip and the cell surface are not well understood, and the disruptive effect of the cantilever tip on cell morphology has not been well characterized. In this study, the disruptive effect of the scanning cantilever tip on cell morphology, in the AFM contact mode, has been investigated. The aims of this study are to identify what kinds of cell morphological changes generally occurred under normal AFM imaging conditions and to find out how long cells remain viable during scanning. Two cell lines, SK-N-SH (human neuroblastoma cells) and AV12 (Syrian hamster cells) were studied in the experiment because these are widely used in biomedical research as an expression system for studying cellular functions of neuronal receptors. The experimental results suggest that the sensitivity of cells to the cantilever disruptive effect is dependent on cell type and that there are patterns observed in the changes of cell morphology induced by the cantilever force in these two cell lines.     

Hydrodynamic damping of a magnetically oscillated cantilever close to a surface

We studied the frequency response of a magnetically driven atomic force microscope (AFM) cantilever close to a sample surface in liquids. Amplitude–frequency (tuning) curves showed pronounced differences in dependence on the tip–sample separation (from 1 to 50 μm), with significant shifts of the resonance peak. A model was developed in which the cantilever was described in a full shape manner and the hydrodynamic forces acting on the cantilever were approximately calculated. The slight inclination of the cantilever to the surface (15°) leads to a force profile along the cantilever. Therefore, the mathematical problem can be strictly solved only numerically. For an approximate analytical solution, the hydrodynamic force profile was approximated by a constant force along the cantilever for large separations and by a point force acting on the tip of the cantilever for small separations. The theoretical results calculated within this model agreed well with the experimental data and allowed to determine the cantilever mass in liquid M^*, the joint mass at the tip end m_t^*, and the coefficient of viscous interaction of the cantilever with free liquid, γ_∞.