Calibration of the normal spring constant of microcantilevers in a parallel fluid flow

Contact-free experimental determination of the static flexural spring constant of cantilever sensors using a microfluidic force tool

Micro- and nanocantilevers are employed in atomic force microscopy (AFM) and in micro- and nanoelectromechanical systems (MEMS and NEMS) as sensing elements. They enable nanomechanical measurements, are essential for the characterization of nanomaterials, and form an integral part of many nanoscale devices. Despite the fact that numerous methods described in the literature can be applied to determine the static flexural spring constant of micro- and nanocantilever sensors, experimental techniques that do not require contact between the sensor and a surface at some point during the calibration process are still the exception rather than the rule. We describe a noncontact method using a microfluidic force tool that produces accurate forces and demonstrate that this, in combination with a thermal noise spectrum, can provide the static flexural spring constant for cantilever sensors of different geometric shapes over a wide range of spring constant values (≈0.8-160 N/m).

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%.

Determination of the spring constants of the higher flexural modes of microcantilever sensors

A method for the simultaneous calibration of the spring constants of all flexural modes of microcantilevers is presented. It is based on a flow of gas from a microchannel that interacts with the microcantilever. The gas flow causes a measurable shift in the resonance frequencies of thermal noise spectra of the flexural modes. From the magnitude of the frequency shifts of the individual modes the spring constants can be determined with high accuracy and precision. The method is non-invasive and does not risk damage to the cantilever. Experimental data are presented for several V-shaped and rectangular cantilevers with nominal fundamental spring constants in the range of 0.03-1.75 N m(-1). The spring constants of the fundamental modes compare favorably to those obtained using the Sader method. The higher modes of oscillation are readily calibrated with experimental uncertainties of 5-10%.

A method for atomic force microscopy cantilever stiffness calibration under heavy fluid loading

This work presents a method for force calibration of rectangular atomic force microscopy (AFM) microcantilevers under heavy fluid loading. Theoretical modeling of the thermal response of microcantilevers is discussed including a fluid-structure interaction model of the cantilever-fluid system that incorporates the results of the fluctuation-dissipation theorem. This model is curve fit to the measured thermal response of a cantilever in de-ionized water and a cost function is used to quantify the difference between the theoretical model and measured data. The curve fit is performed in a way that restricts the search space to parameters that reflect heavy fluid loading conditions. The resulting fitting parameters are used to calibrate the cantilever. For comparison, cantilevers are calibrated using Sader's method in air and the thermal noise method in both air and water. For a set of eight cantilevers ranging in stiffness from 0.050 to 5.8 N/m, the maximum difference between Sader's calibration performed in air and the new method performed in water was 9.4%. A set of three cantilevers that violate the aspect ratio assumption associated with the fluid loading model (length-to-width ratios less than 3.5) ranged in stiffness from 0.85 to 4.7 N/m and yielded differences as high as 17.8%.

Hydrodynamic methods for calibrating the normal spring constant of microcantilevers

Knowledge of the spring constants of microcantilevers is vital in atomic force microscopy and for cantilever-based devices that are, for example, employed as probes in biomedical applications. We compare two recently developed hydrodynamic methods for the determination of the normal spring constant of microcantilevers. Both approaches are non-invasive when determining the spring constant and require only knowledge of the thermal noise response of the cantilever in a fluid and its plan view dimensions. The methods do not bear the risk of damaging the cantilever and are therefore attractive for example in mass sensing applications in cases where the cantilever has been modified, e.g. with a coating. The specific strengths of the methods are discussed and the results for a variety of cantilevers are presented and compared.

Atomic force microscope cantilever spring constant evaluation for higher mode oscillations: a kinetostatic method

Our previous study of the particle mass sensor has shown a large ratio (up to thousands) between the spring constants of a rectangular cantilever in higher mode vibration and at the static bending or natural mode vibration. This has been proven by us through the derived nodal point position equation. That solution is good for a cantilever with the free end in noncontact regime and the probe shifted from the end to an effective section and contacting a soft object. Our further research shows that the same nodal position equation with the proper frequency equations may be used for the same spring constant ratio estimation if the vibrating at higher mode cantilever's free end has a significant additional mass clamped to it or that end is in permanent contact with an elastic or hard measurand object (reference cantilever). However, in the latter case, the spring constant ratio is much smaller (in tens) than in other mentioned cases at equal higher (up to fourth) vibration modes. We also present the spring constant ratio for a vibrating at higher eigenmode V-shaped cantilever, which is now in wide use for atomic force microscopy. The received results on the spring constant ratio are in good (within a few percent) agreement with the theoretical and experimental data published by other researchers. The knowledge of a possible spring constant transformation is important for the proper calibration and use of an atomic force microscope with vibrating cantilever in the higher eigenmodes for measurement and imaging with enlarged resolution.

Cantilever Micro-rheometer for the Characterization of Sugar Solutions

The volume required for the rheological characterization of fluids can be minimized by using micromechanical cantilevers as viscosity sensors. Here, a simple measurement tool for the characterization of sugar solutions is proposed. The sensor consists of a micromechanical cantilever as used in an atomic force microscopy which is integrated into a closed fluid handling system. Fluid properties are derived from an analysis of the power spectral density of the fluctuations of the cantilever deflection signal. The data acquisition system is operated with standard consumer computer components, which limits the costs for the hardware. Measurements with different sugar solutions indicate that the sensor system provides reliable viscosity values for sugar concentrations as they occur in biological systems. The viscosities of the sugar solutions could be evaluated with an error smaller than 5 %.