Spring Constant Of Cantilever Research Articles

Static and dynamic calibration of torsional spring constants of cantilevers.

A method for calibrating the dynamic torsional spring constant of cantilevers by directly measuring the thermally driven motion of the cantilever with an interferometer is presented. Random errors in calibration were made negligible (<1%) by averaging over multiple measurements. The errors in accuracy of ±5% or ±10% for both of the cantilevers calibrated in this study were limited only by the accuracy of the laser Doppler vibrometer (LDV) used to measure thermal fluctuations. This is a significant improvement over commonly used methods that result in large and untraceable errors resulting from assumptions made about the cantilever geometry, material properties, and/or hydrodynamic physics of the surroundings. Subsequently, the static torsional spring constant is determined from its dynamic counterpart after careful LDV measurements of the torsional mode shape, backed by finite element analysis simulations. A meticulously calibrated cantilever is used in a friction force microscopy experiment that measures the friction difference and interfacial shear strength (ISS) between graphene and a silicon dioxide AFM probe. Accurate calibration can resolve discrepancies between different experimental methods, which have contributed to a large scatter in the reported friction and ISS values in the literature to date.

An ultra-precision tool nanoindentation instrument for replication of single point diamond tool cutting edges

Precision replication of the diamond tool cutting edge is required for non-destructive tool metrology. This paper presents an ultra-precision tool nanoindentation instrument designed and constructed for replication of the cutting edge of a single point diamond tool onto a selected soft metal workpiece by precisely indenting the tool cutting edge into the workpiece surface. The instrument has the ability to control the indentation depth with a nanometric resolution, enabling the replication of tool cutting edges with high precision. The motion of the diamond tool along the indentation direction is controlled by the piezoelectric actuator of a fast tool servo (FTS). An integrated capacitive sensor of the FTS is employed to detect the displacement of the diamond tool. The soft metal workpiece is attached to an aluminum cantilever whose deflection is monitored by another capacitive sensor, referred to as an outside capacitive sensor. The indentation force and depth can be accurately evaluated from the diamond tool displacement, the cantilever deflection and the cantilever spring constant. Experiments were carried out by replicating the cutting edge of a single point diamond tool with a nose radius of 2.0 mm on a copper workpiece surface. The profile of the replicated tool cutting edge was measured using an atomic force microscope (AFM). The effectiveness of the instrument in precision replication of diamond tool cutting edges is well-verified by the experimental results.

Phase shifter tuned by varying the spring constant of a nanomechanical cantilever

This paper presents a theoretical study of an adaptive microwave phase shifter based on a nanomechanical resonator. The phase shifter can be tuned to provide an arbitrary phase rotation in the range between –90° and 90° by adjusting the bias voltage applied to the resonator. An analytical model is developed to reveal the underlying mechanism of the phase shifter. It is also analytically demonstrated that the operating frequency can be selected by adjusting the bias voltage. In addition, it is found that the bandwidth is tunable via the quality factor for the cantilever.

Investigation of a small force standard with the mass based method

&lt;p class="Abstract"&gt;&lt;span lang="EN-US"&gt;A small force standard with the mass based method &lt;/span&gt;&lt;span lang="EN-US"&gt;i&lt;/span&gt;&lt;span lang="EN-US"&gt;s developed. It is composed of an electromagnetic compensation balance, a nano-positioning stage and a displacement and pitch angle adjustment unit. The small force standard is used to measure spring constant&lt;/span&gt;&lt;span lang="EN-US"&gt;s&lt;/span&gt;&lt;span lang="EN-US"&gt; of cantilevers in Atomic Force Microscope (AFM) and forces of small force transducers in related applications. Since the balance and the nano-positioning stage are traced to mass and length standards, respectively, the small force standard is SI traceable. The principle and structure of the small force standard and its main components are discussed in detail. The spring constants of several cantilevers used in AFM were calibrated by the small force standard. The calibration results are demonstrated and uncertainties are evaluated. For verifying metrological characteristics of the small force standard, a preliminary international comparison between NIM, PTB and KRISS was carried out. The comparison results &lt;/span&gt;&lt;span lang="EN-US"&gt;wer&lt;/span&gt;&lt;span lang="EN-US"&gt;e in good agreement and indicate that the calibration capabilities of the participating small force standards &lt;/span&gt;&lt;span lang="EN-US"&gt;wer&lt;/span&gt;&lt;span lang="EN-US"&gt;e equivalent within their reported uncertainties.&lt;/span&gt;&lt;/p&gt;

A virtual instrument to standardise the calibration of atomic force microscope cantilevers.

Atomic force microscope (AFM) users often calibrate the spring constants of cantilevers using functionality built into individual instruments. This calibration is performed without reference to a global standard, hindering the robust comparison of force measurements reported by different laboratories. Here, we describe a virtual instrument (an internet-based initiative) whereby users from all laboratories can instantly and quantitatively compare their calibration measurements to those of others-standardising AFM force measurements-and simultaneously enabling non-invasive calibration of AFM cantilevers of any geometry. This global calibration initiative requires no additional instrumentation or data processing on the part of the user. It utilises a single website where users upload currently available data. A proof-of-principle demonstration of this initiative is presented using measured data from five independent laboratories across three countries, which also allows for an assessment of current calibration.

A technique for determining Young's modulus of biological objects using atomic-force microscopy in the wide temperature range below Rt

In the paper, we suggest widening the scope of cryogenic atomic-force microscopy in order to determine Young's modulus of biological objects over a temperature range stretching from 30 to 300K. A new technique assumes some modification of the arrangement for the optical system recording the probe position in the cryogenic atomic-force microscope and making measurements of approach and retraction force-distance curves under these conditions. Young's modulus is determined from the obtained data by calculation taking into account the temperature dependence of a cantilever spring constant. As an example we have obtained a temperature dependence of Young's modulus for polylysine in the temperature range stretching from 60 to 300K (RT) using the proposed technique.

SI-traceable determination of the spring constant of a soft cantilever using the nanonewton force facility based on electrostatic methods

The PTB’s (Physikalisch-Technische Bundesanstalt, Germany) nanonewton force facility, first presented in work by Nesterov (2007 Meas. Sci. Technol. 18 360–6), Nesterov (2009 Meas. Sci. Technol. 20 084012) and Nesterov et al (2009 Metrologia 46 277–82), has been significantly improved and used to measure the stiffness of a cantilever. The facility is based on a disc pendulum with electrostatic reduction of its deflection and stiffness. In this paper, we will demonstrate that the facility is able to measure horizontal forces in the range below 1 μN with a resolution below 5 pN and an uncertainty below 2.7% for a measured force of 1 nN at a measurement duration of about 20 s. We will demonstrate the possibility of using this facility as a calibration device that can accurately determine spring constants of soft cantilevers (K ≲ 0.1 N m−1) with traceability to the SI units. The method and the results of measuring the spring constant of a soft cantilever (K = 0.125 N m−1) in air, in a medium vacuum, in a high vacuum and in nitrogen are presented. We will show that a relative standard uncertainty of the spring constant calibration of better than 0.3% (measurement in a medium vacuum) and a repeatability of better than 0.04% are achieved.

Mechanical properties and applications of custom-built gold AFM cantilevers

Silicon and silicon nitride metal-coated cantilevers have been in use for a long time in several scanning probe microscopy applications that require electrically conductive probes. However, conventional metalized cantilevers present several drawbacks such as limited life-time of the metal layer due to wear, and increased tip radius. This work focuses on monolithic metallic cantilevers developed in order to overcome the limitations of conventional metalized atomic force microscopy (AFM) probes. These custom-made cantilevers are designed for several applications such as in current-sensing AFM (CSAFM), Kelvin probe force microscopy (KPFM), and tip-enhanced Raman spectroscopy (TERS). Determination of the dynamic and static mechanical properties of these cantilevers in a non-destructive way is reported here. Key parameters under investigation are the cantilever spring constant and the frequency response using finite element method (FEM). Gold cantilevers are selected for this study, which allow optimizing the design and the process of developing these metallic cantilevers with parameters engineered for the applications mentioned above. This work contributes to the establishment and applicability of custom-made probes in advanced scanning probe microscopy methods and their performance understanding using computer simulations.

Excluding Contact Electrification in Surface Potential Measurement Using Kelvin Probe Force Microscopy.

Kelvin probe force microscopy (KPFM), a characterization method that could image surface potentials of materials at the nanoscale, has extensive applications in characterizing the electric and electronic properties of metal, semiconductor, and insulator materials. However, it requires deep understanding of the physics of the measuring process and being able to rule out factors that may cause artifacts to obtain accurate results. In the most commonly used dual-pass KPFM, the probe works in tapping mode to obtain surface topography information in a first pass before lifting to a certain height to measure the surface potential. In this paper, we have demonstrated that the tapping-mode topography scan pass during the typical dual-pass KPFM measurement may trigger contact electrification between the probe and the sample, which leads to a charged sample surface and thus can introduce a significant error to the surface potential measurement. Contact electrification will happen when the probe enters into the repulsive force regime of a tip-sample interaction, and this can be detected by the phase shift of the probe vibration. In addition, the influences of scanning parameters, sample properties, and the probe's attributes have also been examined, in which lower free cantilever vibration amplitude, larger adhesion between the probe tip and the sample, and lower cantilever spring constant of the probe are less likely to trigger contact electrification. Finally, we have put forward a guideline to rationally decouple contact electrification from the surface potential measurement. They are decreasing the free amplitude, increasing the set-point amplitude, and using probes with a lower spring constant.

Calibration of effective spring constants of colloidal probes using reference cantilever method

Abstract The precise measurement of the effective spring constant of a colloidal probe's cantilever is very important for the accurate application of a desired contact load to a sample surface since the contact load is determined by multiplying the effective spring constant of the cantilever and its deflection. This paper presents a method of directly measuring the effective spring constant of a colloidal probe by the reference cantilever method. The distinctive feature of the reference cantilever method in this study is that the reference cantilever has a tip. The tip determines the location of the loading point on the reference cantilever and enables detecting the loading point on the colloidal probe's cantilever by monitoring the change in a force distance curve depending on the contact point between a spherical particle and the tip. The effective spring constants of the colloidal probes measured by the proposed method are compared to those determined by finite element analysis (FEA). The importance of calibrating the effective spring constant of the colloidal probe in applied forces and mechanical tests is investigated by comparing force distance curves and nanometer-scale wear test results obtained at two different spring constant settings.

Magnetically Actuated Single-Walled Carbon Nanotubes.

We couple magnetic tweezer techniques with standard lithography methods to make magnetically actuated single-walled carbon nanotube (SWNT) devices. Parallel arrays of 4-10 μm-long SWNT cantilevers are patterned with one end anchored to the substrate and the other end attached to a micron-scale iron magnetic tag that is free to move in solution. Thermal fluctuations of this tag provide a direct measurement of the spring constant of the SWNT cantilevers, yielding values of 10(-7)-10(-8) N/m. This tag is also a handle for applying forces and torques using externally applied magnetic field gradients. These techniques provide a platform on which interaction forces between SWNTs and other objects such as biomolecules and cells can be measured in situ.

Calibration of the effective spring constant of ultra-short cantilevers for a high-speed atomic force microscope

Ultra-short cantilevers are a new type of cantilever designed for the next generation of high-speed atomic force microscope (HS-AFM). Ultra-short cantilevers have smaller dimensions and higher resonant frequency than conventional AFM cantilevers. Moreover, their geometry may also be different from the conventional beam-shape or V-shape. These changes increase the difficulty of determining the spring constant for ultra-short cantilevers, and hence limit the accuracy and precision of force measurement based on a HS-AFM. This paper presents an experimental method to calibrate the effective spring constant of ultra-short cantilevers. By using a home-made AFM head, the cantilever is bent against an electromagnetic compensation balance under servo control. Meanwhile the bending force and the cantilever deflection are synchronously measured by the balance and the optical lever in the AFM head, respectively. Then the effective spring constant is simply determined as the ratio of the force to the corresponding deflection. Four ultra-short trapezoid shape cantilevers were calibrated using this method. A quantitative uncertainty analysis showed that the combined relative standard uncertainty of the calibration result is less than 2%, which is better than the uncertainty of any previously reported techniques.

Note: Improved calibration of atomic force microscope cantilevers using multiple reference cantilevers.

Overall precision of the simplified calibration method in J. E. Sader et al., Rev. Sci. Instrum. 83, 103705 (2012), Sec. III D, is dominated by the spring constant of the reference cantilever. The question arises: How does one take measurements from multiple reference cantilevers, and combine these results, to improve uncertainty of the reference cantilever's spring constant and hence the overall precision of the method? This question is addressed in this note. Its answer enables manufacturers to specify of a single set of data for the spring constant, resonant frequency, and quality factor, from measurements on multiple reference cantilevers. With this data set, users can trivially calibrate cantilevers of the same type.

Precise and direct method for the measurement of the torsion spring constant of the atomic force microscopy cantilevers.

A direct method for the evaluation of the torsional spring constants of the atomic force microscope cantilevers is presented in this paper. The method uses a nanoindenter to apply forces at the long axis of the cantilever and in the certain distance from it. The torque vs torsion relation is then evaluated by the comparison of the results of the indentations experiments at different positions on the cantilever. Next, this relation is used for the precise determination of the torsional spring constant of the cantilever. The statistical analysis shows that the standard deviation of the calibration measurements is equal to approximately 1%. Furthermore, a simple method for calibration of the photodetector's lateral response is proposed. The overall procedure of the lateral calibration constant determination has the accuracy approximately equal to 10%.

Measurement of cantilever spring constant using an electrostatic sensing and actuating force measurement system

We report the measurement of the spring constant of a cantilever using an electrostatic sensing and actuating force measurement system. The force measurement system consists of three main parts: a monolithic flexure stage, a three-electrode capacitor, and a digital controller. The most important feature of the system is the utilization of the three-electrode capacitor simultaneously as a position sensor and an electrostatic force generator. This allows us to constantly adjust and maintain gap distances between three electrodes, which is critical for this design to generate precise electrostatic compensation force. The force measurement system was partly upgraded and its performance was improved. The uncertainty estimation for the upgraded force measurement system is given. By measuring the spring constant of a commercially available cantilever force sensor, we demonstrate that the force measurement system is capable of calibrating the cantilever spring constant. The spring constant of the cantilever force sensor was measured to be (2227 ± 63) mN m−1 (coverage factor k = 2), corresponding to a relative expanded uncertainty of 2.8%.

Characterizing the free and surface-coupled vibrations of heated-tip atomic force microscope cantilevers**Publication of NIST, an agency of the US government, not subject to copyright.

Combining heated-tip atomic force microscopy (HT-AFM) with quantitative methods for determining surface mechanical properties, such as contact resonance force microscopy, creates an avenue for nanoscale thermomechanical property characterization. For nanomechanical methods that employ an atomic force microscope cantilever’s vibrational modes, it is essential to understand how the vibrations of the U-shaped HT-AFM cantilever differ from those of a more traditional rectangular lever, for which analytical techniques are better developed. Here we show, with a combination of finite element analysis (FEA) and experiments, that the HT-AFM cantilever exhibits many more readily-excited vibrational modes over typical AFM frequencies compared to a rectangular cantilever. The arms of U-shaped HT-AFM cantilevers exhibit two distinct forms of flexural vibrations that differ depending on whether the two arms are vibrating in-phase or out-of-phase with one another. The in-phase vibrations are qualitatively similar to flexural vibrations in rectangular cantilevers and generally show larger sensitivity to surface stiffness changes than the out-of-phase vibrations. Vibration types can be identified from their frequency and by considering vibration amplitudes in the horizontal and vertical channels of the AFM at different laser spot positions on the cantilever. For identifying contact resonance vibrational modes, we also consider the sensitivity of the resonant frequencies to a change in applied force and hence to tip–sample contact stiffness. Finally, we assess how existing analytical models can be used to accurately predict contact stiffness from contact-resonance HT-AFM results. A simple two-parameter Euler–Bernoulli beam model provided good agreement with FEA for in-phase modes up to a contact stiffness 500 times the cantilever spring constant. By providing insight into cantilever vibrations and exploring the potential of current analysis techniques, our results lay the groundwork for future use of HT-AFM cantilevers for accurate nanomechanical property measurements.

Spring constant calibration techniques for next-generation fast-scanning atomic force microscope cantilevers

As a recent technological development, high-speed atomic force microscopy (AFM) has provided unprecedented insights into dynamic processes on the nanoscale, and is capable of measuring material property variation over short timescales. Miniaturized cantilevers developed specifically for high-speed AFM differ greatly from standard cantilevers both in size and dynamic properties, and calibration of the cantilever spring constant is critical for accurate, quantitative measurement. This work investigates specifically, the calibration of these new-generation cantilevers for the first time. Existing techniques are tested and the challenges encountered are reported and the most effective approaches for calibrating fast-scanning cantilevers with high accuracy are identified, providing a resource for microscopists in this rapidly developing field. Not only do these cantilevers offer faster acquisition of images and force data but due to their high resonant frequencies (up to 2 MHz) they are also excellent mass sensors. Accurate measurement of deposited mass requires accurate calibration of the cantilever spring constant, therefore the results of this work will also be useful for mass-sensing applications.

Characterization of the interaction between AFM tips and surface nanobubbles.

While the presence of gaseous enclosures observed at various solid-water interfaces, the so-called "surface nanobubles", has been confirmed by many groups in recent years, their formation, properties, and stability have not been convincingly and exhaustively explained. Here we report on an atomic force microscopy (AFM) study of argon nanobubbles on highly oriented pyrolitic graphite (HOPG) in water to elucidate the properties of nanobubble surfaces and the mechanism of AFM tip-nanobubble interaction. In particular, the deformation of the nanobubble-water interface by the AFM tip and the question whether the AFM tip penetrates the nanobubble during scanning were addressed by this combined intermittent contact (tapping) mode and force volume AFM study. We found that the stiffness of nanobubbles was smaller than the cantilever spring constant and comparable with the surface tension of water. The interaction with the AFM tip resulted in severe quasi-linear deformation of the bubbles; however, in the case of tip-bubble attraction, the interface deformed toward the tip. We tested two models of tip-bubble interaction, namely, the capillary force and the dynamic interaction model, and found, depending on the tip properties, good agreement with experimental data. The results showed that the tip-bubble interaction strength and the magnitude of the bubble deformation depend strongly on tip and bubble geometry and on tip and substrate material, and are very sensitive to the presence of contaminations that alter the interfacial tension. In particular, nanobubbles interacted differently with hydrophilic and hydrophobic AFM tips, which resulted in qualitatively and quantitatively different force curves measured on the bubbles in the experiments. To minimize bubble deformation and obtain reliable AFM results, nanobubbles must be measured with a sharp hydrophilic tip and with a cantilever having a very low spring constant in a contamination-free system.

AFM nanoscale indentation in air of polymeric and hybrid materials with highly different stiffness

Abstract In this study, nanomechanical properties of a variety of polymeric materials was investigated by means of AFM. In particular, selecting different AFM probes, poly(methyl methacrylate) (PMMA), polydimethylsiloxane (PDMS) bulk samples, sol–gel hybrid thin films and hydrated hyaluronic acid hydrogels were indented in air to determine the elastic modulus. The force–distance curves and the indentation data were found to be greatly affected by the cantilever stiffness and by tip geometry. AFM indentation tests show that the choice of the cantilever spring constant and of tip shape is crucially influenced by elastic properties of samples. When adhesion-dominated interactions occur between the tip and the surface of samples, force–displacement curves reveal that a suitable functionalization of AFM probes allows the control of such interactions and the extraction of Young’ modulus from AFM curves that would be otherwise unfeasible. By applying different mathematical models depending on AFM probes and materials under investigation, the values of Young's modulus were obtained and compared to those measured by rheological and dynamic mechanical analysis or to literature data. Our results show that a wide range of elastic moduli (10 kPa–10 GPa) can be determined by AFM in good agreement with those measured by conventional macroscopic measurements.

Calibration of the torsional and lateral spring constants of cantilever sensors

A method suitable for the calibration of the spring constants of all torsional and lateral eigenmodes of micro- and nanocantilever sensors is described. Such sensors enable nanomechanical measurements and the characterization of nanomaterials, for example with atomic force microscopy. The method presented involves the interaction of a flow of fluid from a microchannel with the cantilever beam. Forces imparted by the flow cause the cantilever to bend and induce a measurable change of the torsional and lateral resonance frequencies. From the frequency shifts the cantilever spring constants can be determined. The method does not involve physical contact between the cantilever or its tip and a hard surface. As such it is non-invasive and does not risk damage to the cantilever. Experimental data is presented for two rectangular microcantilevers with fundamental flexural spring constants of 0.046 and 0.154 N m−1. The experimentally determined torsional stiffness values are compared with those obtained by the Sader method. We demonstrate that the torsional spring constants can be readily calibrated using the method with an accuracy of around 15%.