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.     

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.     

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.     

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.     

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.     

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.     

Kinetostatic model of spring constant ratios for an AFM cantilever with end extended mass

In previous work we showed that the kinetostatic method is very effective in computing the increase in value of the spring constants of an AFM free (with or without added mass) and supported rectangular cantilever for higher mode oscillations relative to their values for natural vibration. We have considered in all previous cases that added mass is a concentrated one. However, the additional mass may be an extended one particularly in the case of a V-shaped cantilever. In this article we consider the influence of the constituent beam’s (leg’s) mutual skew and the altered position of the nodal points in the case when the attached extended triangular (trapezoid) mass of the V-shaped cantilever has a significant moment of rotational inertia and a center of this mass gravity located beyond the constituent beam end. We show that considering these effects in using the kinetostatic model yields results for the ratios of the spring constants at higher modes of oscillation and their values at the first frequency natural vibration for a V-shaped cantilever which are in good agreement with the thermomechanical noise amplitudes obtained by other researchers. This should prove helpful for the proper calibration of V-shaped cantilevers whose application with higher modes oscillation provides increased measurement sensitivity.     

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.     

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.     

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.     

The effect of off-end tip distance on the nanomanipulation based on rectangular and V-shape cantilevered AFMs

The AFM system, which is used as a nanomanipulator, includes a probe consistent of a cantilever and a tapered tip. In cantilevers, the tip can be located in different distances from the cantilever free end. This causes to change in stiffness of the cantilever and therefore changing in pushing force of the nanomanipulation. In this paper, the effect of the tip distance on the cantilever stiffness is studied using the equations of Hazel, and Neumeister and Ducker (ND), and a new equation to correct the torsional stiffness of V-shaped cantilevers (VSC) is proposed, which is based on the ND equation. Then, the effect of distance on pushing force of AFM-based nanomanipulations with rectangular cantilevered (RC) and VSC AFMs is simulated. The obtained results using proposed equation show that increasing of distance causes to non-linear increment of torsional stiffness of VSC. Error of the proposed equation is achieved less than 3% in comparison with result of torsional stiffness equation of ND. Moreover, it is observed that the torsional stiffness of VSC predicted by Hazel’s equation is considerably inaccurate. In nanomanipulation studies, the necessary pushing forces of nanoparticle motion are increased by increment of distance, for both types of cantilevers (RC and VSC). Moreover, critical time for RC AFM increases, but in the case of VSC AFM, the critical time decreases at first, then it is almost constant at a limited range of d, and finally it starts to increase by increasing the distance.     

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.     

Direct measurement of cantilever spring constants and correction for cantilever irregularities using an instrumented indenter

A method is presented that allows direct measurement of a wide range of spring constants of cantilevers using an indentation instrument with an integrated optical microscope. An uncertainty of less than 10% can be achieved for spring constants from 0.1 to 10(2) Nm. The technique makes it possible to measure the spring constant at any desired location on a cantilever of any shape, particularly at the tip location of an atomic force microscopy cantilever. The article also demonstrates a technique to detect and correct apparent length anomalies of cantilevers by analyzing spring constants at multiple positions.     

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.     

Finite-element vibration analysis of tapping-mode atomic force microscopy in liquid

Investigation of morphology and mechanical properties of biological specimens using atomic force microscopy (AFM) often requires its operation in liquid environment. Due to the hydrodynamic force, the vibration of AFM cantilevers in liquid shows dramatically different dynamic characteristics from that in air. A good understanding of the dynamics of AFM cantilevers vibrating in liquid is needed for the interpretation of scanning images, selection of AFM operating conditions, and evaluation of sample's mechanical properties. In this study, a finite element (FE) model is used for frequency and transient response analysis of AFM cantilevers in tapping mode (TM) operated in air or liquid. Hydrodynamic force exerted by the fluid on AFM cantilevers is approximated by additional mass and hydrodynamic damping. The additional mass and hydrodynamic damping matrices corresponding to beam elements are derived. With this model, numerical simulations are performed for an AFM cantilever to obtain the frequency and transient responses of the cantilever in air and liquid. The comparison between our simulated results and the experimentally obtained ones shows good agreement. Based on the simulations, different characteristics of cantilever dynamics in air and liquid are discussed.     

Finite element calculations and fabrication of cantilever sensors for nanoscale detection

Finite element analysis (FEA) is used to study the effect of geometric variations on the properties of rectangular cantilevers and U-shaped Joule-heated cantilevers. Simulations of locally thinned cantilevers as well as of cantilevers modified by the implementing of a hole or a side cut are compared with fabricated cantilevers, which are tuned by focused ion beam (FIB) milling. By locally thinning the cantilevers, the resonance frequency and the spring constant are reduced. For a hole, the internal stress is increased while for a side cut, the lateral spring constant is decreased. Good agreement between the measured and the simulated resonance frequencies is observed. Simulations of the current density and the temperature distributions attained during the passage of current through a doped silicon layer are performed to optimize the design of Joule-heated cantilevers (U-shaped) for thermal gravimetric applications. A very uniform temperature distribution over a region near the apex can be realized by slitting the U-shaped cantilever. In such a way, the heating power can be minimized by effecting only a small variation in the geometry of a U-shaped cantilever. A simple fabrication process for the fabrication of Joule-heated cantilevers is presented, which consists mainly of a uniform conductive p-doped layer.     

Thermomechanical noise of a free v-shaped cantilever for atomic-force microscopy

We have calculated the thermal noise of a v-shaped AFM cantilever (Microlever, Type E, Thermomicroscopes) by means of a finite element analysis. The modal shapes of the first 10 eigenmodes are displayed as well as the numerical constants, which are needed for the calibration using the thermal noise method. In the first eigenmode, values for the thermomechanical noise of the z-displacement at 22 degrees C temperature of square root of u2(1) = A/square root of c(cant) and the photodiode signal (normal-force) of S2(1) = A/square root of c(cant) were obtained. The results also indicate a systematic deviation ofthe spectral density of the thermomechanical noise of v-shaped cantilevers as compared to rectangular beam-shaped cantilevers.     

Improved parallel scan method for nanofriction force measurement with atomic force microscopy

Based on Ruan and Bhushan's study [J. Ruan and B. Bhushan, J. Tribol. 116, 378 (1994)], an improved method for quantitative nano/microfriction force measurements with the atomic force microscope (AFM) is presented. The related theoretical derivation is given in detail. The coefficient of friction can be calculated by scanning in the direction parallel to the long axis of the AFM cantilever. Then conversion factor, which can convert the lateral deflection response of the photodetector into corresponding friction force, is identified with the Meyer and Amer method [G. Meyer and N. M. Ame, Appl. Phys. Lett. 57, 2089 (1990)]. Like Ruan and Bhushan method, the advantage of this approach is that the coefficient of friction can be obtained with the plan-view geometry of AFM cantilevers and some common uncertainties, such as thickness, coating, and material properties, are not necessary. The result of the experiments performed utilizing rectangular cantilevers of different lengths shows that this improved method produces an accurate agreement for cantilevers of different lengths, thus the method can be used to measure nano/microfriction force.     

Study of thermal and acoustic noise interferences in low stiffness atomic force microscope cantilevers and characterization of their dynamic properties

The atomic force microscope (AFM) is a powerful tool for the measurement of forces at the micro/nano scale when calibrated cantilevers are used. Besides many existing calibration techniques, the thermal calibration is one of the simplest and fastest methods for the dynamic characterization of an AFM cantilever. This method is efficient provided that the Brownian motion (thermal noise) is the most important source of excitation during the calibration process. Otherwise, the value of spring constant is underestimated. This paper investigates noise interference ranges in low stiffness AFM cantilevers taking into account thermal fluctuations and acoustic pressures as two main sources of noise. As a result, a preliminary knowledge about the conditions in which thermal fluctuations and acoustic pressures have closely the same effect on the AFM cantilever (noise interference) is provided with both theoretical and experimental arguments. Consequently, beyond the noise interference range, commercial low stiffness AFM cantilevers are calibrated in two ways: using the thermal noise (in a wide temperature range) and acoustic pressures generated by a loudspeaker. We then demonstrate that acoustic noises can also be used for an efficient characterization and calibration of low stiffness AFM cantilevers. The accuracy of the acoustic characterization is evaluated by comparison with results from the thermal calibration.     

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.