Spring Constant Calibration Research Articles

Hydrodynamic function and spring constant calibration of FluidFM micropipette cantilevers

Fluidic force microscopy (FluidFM) fuses the force sensitivity of atomic force microscopy with the manipulation capabilities of microfluidics by using microfabricated cantilevers with embedded fluidic channels. This innovation initiated new research and development directions in biology, biophysics, and material science. To acquire reliable and reproducible data, the calibration of the force sensor is crucial. Importantly, the hollow FluidFM cantilevers contain a row of parallel pillars inside a rectangular beam. The precise spring constant calibration of the internally structured cantilever is far from trivial, and existing methods generally assume simplifications that are not applicable to these special types of cantilevers. In addition, the Sader method, which is currently implemented by the FluidFM community, relies on the precise measurement of the quality factor, which renders the calibration of the spring constant sensitive to noise. In this study, the hydrodynamic function of these special types of hollow cantilevers was experimentally determined with different instruments. Based on the hydrodynamic function, a novel spring constant calibration method was adapted, which relied only on the two resonance frequencies of the cantilever, measured in air and in a liquid. Based on these results, our proposed method can be successfully used for the reliable, noise-free calibration of hollow FluidFM cantilevers.

Determination of calibration parameters of cantilevers of arbitrary shape by finite element analysis

The use of atomic force microscopy in nanomechanical measurements requires accurate calibration of the cantilever's spring constant (kc) and the optical lever sensitivity (OLS). The thermal method, based on the cantilever's thermal fluctuations in fluids, allows estimation of kc in a fast, non-invasive mode. However, differences in the cantilever geometry and mounting angle require the knowledge of three correction factors to get a good estimation of kc: the contribution of the oscillation mode to the total amplitude, the shape difference between the free and end-loaded configurations, and the tilt of the cantilever with respect to the measured surface. While the correction factors for traditional rectangular and V-shaped cantilever geometries have been reported, they must be determined for cantilevers with non-traditional geometries and large tips. Here, we develop a method based on finite element analysis to estimate the correction factors of cantilevers with arbitrary geometry and tip dimensions. The method relies on the numerical computation of the effective cantilever mass. The use of the correction factor for rectangular geometries in our model cantilever (PFQNM-LC) will lead to values underestimated by 16%. In contrast, experiments using pre-calibrated cantilevers revealed a maximum uncertainty below 5% in the estimation of the OLS, verifying our approach.

Large colloidal probes for atomic force microscopy: Fabrication and calibration issues.

Atomic force microscopy (AFM) is a powerful tool to investigate interaction forces at the micro and nanoscale. Cantilever stiffness, dimensions and geometry of the tip can be chosen according to the requirements of the specific application, in terms of spatial resolution and force sensitivity. Colloidal probes (CPs), obtained by attaching a spherical particle to a tipless (TL) cantilever, offer several advantages for accurate force measurements: tunable and well-characterisable radius; higher averaging capabilities (at the expense of spatial resolution) and sensitivity to weak interactions; a well-defined interaction geometry (sphere on flat), which allows accurate and reliable data fitting by means of analytical models. The dynamics of standard AFM probes has been widely investigated, and protocols have been developed for the calibration of the cantilever spring constant. Nevertheless, the dynamics of CPs, and in particular of large CPs, with radius well above 10 μm and mass comparable, or larger, than the cantilever mass, is at present still poorly characterized. Here we describe the fabrication and calibration of (large) CPs. We describe and discuss the peculiar dynamical behaviour of CPs, and present an alternative protocol for the accurate calibration of the spring constant.

Traceable Lateral Force Calibration (TLFC) for Atomic Force Microscopy

Efforts to reliably measure AFM lateral forces have been impeded by the difficulties in obtaining appropriate calibration standards, applying those force standards to the apex of the tip, and quantifying calibration uncertainty. Here we propose a new method, Traceable Lateral Force Calibration (TLFC), which combines the reliability of direct methods with the convenience of indirect/semi-direct methods. Like other direct methods, ours comprise three essential steps: (1) fabrication of a spring (the Traceable Reference Lever or TRL); (2) calibration of the TRL spring constant; (3) conversion of measurable TRL deflections into absolute lateral force measurements based on its pre-calibrated spring constant (TLFC method). The TRL device, a simple two-axis cantilever, is easy to design, fabricate, and directly pre-calibrate with a standard laboratory microbalance. Following pre-calibration, the TRL device becomes a convenient absolute standard for AFM lateral force measurements. This paper describes the complete method and demonstrates its primary merits, which include (1) traceability to measurement standards; (2) ease of use by outside user groups; (3) absolute measurement errors 1 N/m normal stiffness); (4) robustness over a wide range of common loads, instruments, probes, and environments. While the method and proof-of-concept devices described in this paper were designed primarily for moderate to high load cantilevers (> 1 N/m), we discuss how a next generation of compliant TRL devices can be used with the TLFC method to reliably calibrate arbitrary AFM cantilevers (< < 1 N/m) and forces.

Relaxing constraints on data acquisition and position detection for trap stiffness calibration in optical tweezers.

Optical tweezers find applications in various fields, ranging from biology to physics. One of the fundamental steps necessary to perform quantitative measurements using trapped particles is the calibration of the tweezer's spring constant. This can be done through power spectral density analysis, from forward scattering detection of the particle's position. In this work we propose and experimentally test simplifications to such measurement procedure, aimed at reducing post-processing of recorded data and dealing with acquisition devices that have frequency-dependent electronic noise. In the same line of simplifying the tweezer setup we also present a knife-edge detection scheme that can substitute standard position sensitive detectors.

Spring constant and sensitivity calibration of FluidFM micropipette cantilevers for force spectroscopy measurements

The fluidic force microscope (FluidFM) can be considered as the nanofluidic extension of the atomic force microscope (AFM). This novel instrument facilitates the experimental procedure and data acquisition of force spectroscopy (FS) and is also used for the determination of single-cell adhesion forces (SCFS) and elasticity. FluidFM uses special probes with an integrated nanochannel inside the cantilevers supported by parallel rows of pillars. However, little is known about how the properties of these hollow cantilevers affect the most important parameters which directly scale the obtained spectroscopic data: the inverse optical lever sensitivity (InvOLS) and the spring constant (k). The precise determination of these parameters during calibration is essential in order to gain reliable, comparable and consistent results with SCFS. Demonstrated by our literature survey, the standard error of previously published SCFS results obtained with FluidFM ranges from 11.8% to 50%. The question arises whether this can be accounted for biological diversity or may be the consequence of improper calibration. Thus the aim of our work was to investigate the calibration accuracy of these parameters and their dependence on: (1) the aperture size (2, 4 and 8 µm) of the hollow micropipette type cantilever; (2) the position of the laser spot on the back of the cantilever; (3) the substrate used for calibration (silicon or polystyrene). It was found that both the obtained InvOLS and spring constant values depend significantly on the position of the laser spot. Apart from the theoretically expectable monotonous increase in InvOLS (from the tip to the base of the cantilever, as functions of the laser spot’s position), we discerned a well-defined and reproducible fluctuation, which can be as high as ±30%, regardless of the used aperture size or substrate. The calibration of spring constant also showed an error in the range of −13/+20%, measured at the first 40 µm of the cantilever. Based on our results a calibration strategy is proposed and the optimal laser position which yields the most reliable spring constant values was determined and found to be on the first pair of pillars. Our proposed method helps in reducing the error introduced via improper calibration and thus increases the reliability of subsequent cell adhesion force or elasticity measurements with FluidFM.

Characterisation of the Material and Mechanical Properties of Atomic Force Microscope Cantilevers with a Plan-View Trapezoidal Geometry

Cantilever devices have found applications in numerous scientific fields and instruments, including the atomic force microscope (AFM), and as sensors to detect a wide range of chemical and biological species. The mechanical properties, in particular, the spring constant of these devices is crucial when quantifying adhesive forces, material properties of surfaces, and in determining deposited mass for sensing applications. A key component in the spring constant of a cantilever is the plan-view shape. In recent years, the trapezoidal plan-view shape has become available since it offers certain advantages to fast-scanning AFM and can improve sensor performance in fluid environments. Euler beam equations relating cantilever stiffness to the cantilever dimensions and Young’s modulus have been proven useful and are used extensively to model cantilever mechanical behaviour and calibrate the spring constant. In this work, we derive a simple correction factor to the Euler beam equation for a beam-shaped cantilever that is applicable to any cantilever with a trapezoidal plan-view shape. This correction factor is based upon previous analytical work and simplifies the application of the previous researchers formula. A correction factor to the spring constant of an AFM cantilever is also required to calculate the torque produced by the tip when it contacts the sample surface, which is also dependent on the plan-view shape. In this work, we also derive a simple expression for the torque for triangular plan-view shaped cantilevers and show that for the current generation of trapezoidal plan-view shaped AFM cantilevers, this will be a good approximation. We shall apply both these correction factors to determine Young’s modulus for a range of trapezoidal-shaped AFM cantilevers, which are specially designed for fast-scanning. These types of AFM probes are much smaller in size when compared to standard AFM probes. In the process of analysing the mechanical properties of these cantilevers, important insights are also gained into their spring constant calibration and dimensional factors that contribute to the variability in their spring constant.

Eliminating the effect of acoustic noise on cantilever spring constant calibration

A common use of atomic force microscopy is quantifying local forces through tip-sample interactions between the probe tip and a sample surface. The accuracy of these measurements depends on the accuracy to which the cantilever spring constant is known. Recent work has demonstrated that the measured spring constant of a cantilever can vary up to a factor of five, even for the exact same cantilever measured by different users on different microscopes. Here, we demonstrate that a standard method for calibrating the spring constant (using oscillations due to thermal energy) is susceptible to ambient acoustic noise, which can alter the result significantly. We demonstrate a step-by-step method to measure the spring constant by actively driving the cantilever to measure the resonance frequency and the quality factor, giving results that are unaffected by acoustic noise. Our method can be performed rapidly on any atomic force microscope without any expensive additional hardware.

A non-destructive method to calibrate the torsional spring constant of atomic force microscope cantilevers in viscous environments

Calibration of the torsional spring constant of atomic force microscopy cantilevers is fundamental to a range of applications, from nanoscale friction and lubrication measurements to the characterization of micro-electromechanical systems and the response of biomolecules to external stimuli. Existing calibration methods are either time consuming and destructive (ex situ static approaches), or rely on models using the frequency and quality factor (Q-factor) of the cantilever torsional resonance as input parameters (in situ dynamical approaches). While in situ approaches are usually preferred for their easy implementation and preservation of the cantilever, their dependence on the torsional resonance Q-factor renders calibration in highly viscous environments challenging. This is problematic, for example, in many nanoscale tribological applications. Here, we propose a calibration method that does not depend on the cantilever torsional Q-factor and show how the cantilever deflection can be converted into a lateral force. The method is tested with six cantilevers of different shapes and material composition and in six fluid media. The derived spring constants are compared with predictions from existing methods, demonstrating a higher precision, in particular, for highly viscous liquids.

New design of the cantilevers for radiation pressure investigations

Radiation pressure – RP phenomenon has drawn much interest since the beginning of the 20th century. There are many technologies linked with it and worth mentioning: cavity optomechanics with optical cooling and small forces amplification, single atom trapping, cantilevers spring constant calibration or high-power laser measurement.Nevertheless, the quantitative measurement of the RP is always influenced by many parasitic effects e.g. photothermal effects, which can appear while the beam is absorbed. The goal of the investigations presented in this paper was to design and manufacture a MEMS tool, the so-called RP cantilever, for the reliable interactions studies (i.e. due to radiation pressure) where the photothermal effect is highly reduced. Besides design and manufacturing process we present a set of experiments showing the properties of the proposed technology and its application possibilities.

Determining the spring constant of arbitrarily shaped cantilevers in viscous environments

Accurate calibration of the flexural spring constant of microcantilevers is crucial for sensing devices, microactuators, and atomic force microscopy (AFM). Existing methods rely on precise knowledge of cantilever geometry, make significant simplifications, or require potentially damaging contact with the sample. Here, we develop a simple equation to calculate the flexural spring constants of arbitrarily shaped cantilevers in fluid. Our approach, verified here with AFM, only requires the measurement of two resonance frequencies of the cantilever in air and in a liquid, with no need for additional input or knowledge about the system. We validate the method with cantilevers of different shapes and compare its predictions with existing models. We also show how the method's accuracy can be considerably improved, especially in more viscous liquids, if the effective width of the cantilever is known. Significantly, the developed equations can be extended to calculate the spring constants of the cantilever's higher eigenmodes.

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.

An analytic model for accurate spring constant calibration of rectangular atomic force microscope cantilevers.

Spring constant calibration of the atomic force microscope (AFM) cantilever is of fundamental importance for quantifying the force between the AFM cantilever tip and the sample. The calibration within the framework of thin plate theory undoubtedly has a higher accuracy and broader scope than that within the well-established beam theory. However, thin plate theory-based accurate analytic determination of the constant has been perceived as an extremely difficult issue. In this paper, we implement the thin plate theory-based analytic modeling for the static behavior of rectangular AFM cantilevers, which reveals that the three-dimensional effect and Poisson effect play important roles in accurate determination of the spring constants. A quantitative scaling law is found that the normalized spring constant depends only on the Poisson’s ratio, normalized dimension and normalized load coordinate. Both the literature and our refined finite element model validate the present results. The developed model is expected to serve as the benchmark for accurate calibration of rectangular AFM cantilevers.

Accurate flexural spring constant calibration of colloid probe cantilevers using scanning laser Doppler vibrometry

Calibration of the flexural spring constant for atomic force microscope (AFM) colloid probe cantilevers provides significant challenges. The presence of a large attached spherical added mass complicates many of the more common calibration techniques such as reference cantilever, Sader, and added mass. Even the most promising option, AFM thermal calibration, can encounter difficulties during the optical lever sensitivity measurement due to strong adhesion and friction between the sphere and a surface. This may cause buckling of the end of the cantilever and hysteresis in the approach–retract curves resulting in increased uncertainty in the calibration. Most recently, a laser Doppler vibrometry thermal method has been used to accurately calibrate the normal spring constant of a wide variety of tipped and tipless commercial cantilevers. This paper describes a variant of the technique, scanning laser Doppler vibrometry, optimized for colloid probe cantilevers and capable of spring constant calibration uncertainties near ±1%.

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.

Accurate calibration and uncertainty estimation of the normal spring constant of various AFM cantilevers.

Measurement of force on a micro- or nano-Newton scale is important when exploring the mechanical properties of materials in the biophysics and nanomechanical fields. The atomic force microscope (AFM) is widely used in microforce measurement. The cantilever probe works as an AFM force sensor, and the spring constant of the cantilever is of great significance to the accuracy of the measurement results. This paper presents a normal spring constant calibration method with the combined use of an electromagnetic balance and a homemade AFM head. When the cantilever presses the balance, its deflection is detected through an optical lever integrated in the AFM head. Meanwhile, the corresponding bending force is recorded by the balance. Then the spring constant can be simply calculated using Hooke’s law. During the calibration, a feedback loop is applied to control the deflection of the cantilever. Errors that may affect the stability of the cantilever could be compensated rapidly. Five types of commercial cantilevers with different shapes, stiffness, and operating modes were chosen to evaluate the performance of our system. Based on the uncertainty analysis, the expanded relative standard uncertainties of the normal spring constant of most measured cantilevers are believed to be better than 2%.

Note: calibration of atomic force microscope cantilevers using only their resonant frequency and quality factor.

A simplified method for calibrating atomic force microscope cantilevers was recently proposed by Sader et al. [Rev. Sci. Instrum. 83, 103705 (2012); Sec. III D] that relies solely on the resonant frequency and quality factor of the cantilever in fluid (typically air). This method eliminates the need to measure the hydrodynamic function of the cantilever, which can be time consuming given the wide range of cantilevers now available. Using laser Doppler vibrometry, we rigorously assess the accuracy of this method for a series of commercially available cantilevers and explore its performance under non-ideal conditions. This shows that the simplified method is highly accurate and can be easily implemented to perform fast, robust, and non-invasive spring constant calibration.

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.

Note: Spring constant calibration of nanosurface-engineered atomic force microscopy cantilevers

The determination of the dynamic spring constant (kd) of atomic force microscopy cantilevers is of crucial importance for converting cantilever deflection to accurate force data. Indeed, the non-destructive, fast, and accurate measurement method of the cantilever dynamic spring constant by Sader et al. [Rev. Sci. Instrum. 83, 103705 (2012)] is confirmed here for plane geometry but surface modified cantilevers. It is found that the measured spring constants (keff, the dynamic one kd), and the calculated (kd,1) are in good agreement within less than 10% error.

Accurate spring constant calibration for very stiff atomic force microscopy cantilevers

There are many atomic force microscopy (AFM) applications that rely on quantifying the force between the AFM cantilever tip and the sample. The AFM does not explicitly measure force, however, so in such cases knowledge of the cantilever stiffness is required. In most cases, the forces of interest are very small, thus compliant cantilevers are used. A number of methods have been developed that are well suited to measuring low stiffness values. However, in some cases a cantilever with much greater stiffness is required. Thus, a direct, traceable method for calibrating very stiff (approximately 200 N/m) cantilevers is presented here. The method uses an instrumented and calibrated nanoindenter to determine the stiffness of a reference cantilever. This reference cantilever is then used to measure the stiffness of a number of AFM test cantilevers. This method is shown to have much smaller uncertainty than previously proposed methods. An example application to fracture testing of nanoscale silicon beam specimens is included.