Normal and torsional spring constants of atomic force microscope cantilevers

Abstract

Two methods commonly used to measure the normal spring constants of atomic force microscope cantilevers are the added mass method of Cleveland et al. ͓J. P. Cleveland et al., Rev. Sci. Instrum. 64, 403 ͑1993͔͒, and the unloaded resonance technique of Sader et al. ͓J. E. Sader, J. W. M. Chon, and P. Mulvaney, Rev. Sci. Instrum. 70, 3967 ͑1999͔͒. The added mass method involves measuring the change in resonant frequency of the fundamental mode of vibration upon the addition of known masses to the free end of the cantilever. In contrast, the unloaded resonance technique requires measurement of the unloaded resonant frequency and quality factor of the fundamental mode of vibration, as well as knowledge of the plan view dimensions of the cantilever and properties of the fluid. In many applications, such as frictional force microscopy, the torsional spring constant is often required. Consequently, in this article, we extend both of these techniques to allow simultaneous calibration of both the normal and torsional spring constants. We also investigate the validity and applicability of the unloaded resonance method when a mass is attached to the free end of the cantilever due to its importance in practice.

Direct measurements of non-ionic attraction and nanoscaled lubrication in biomimetic composites from nanofibrillated cellulose and modified carboxymethylated cellulose

“…Before gluing the cellulose microparticles, the spring constants of the cantilevers were determined from the analysis of the cantilever thermal vibrations with the software AFM Tune IT v2.5 (ForceIT, Sweden) and the use of Sader's equations. 47,48 Values of about 0.25 N/m and 25 1.9×10 -9 Nm/rad were obtained for the normal and lateral spring constants, respectively. CMC-g-PEG was adsorbed either on NFC film, on the cellulose colloidal probe or on both, from 100 mg/L polymer solution at pH 4.5 overnight, followed by rinsing with deionized water and drying under nitrogen flow.…”

Dynamic behaviour in piezoresponse force microscopy

“…(22) The resonant frequencies for the electrostatically driven free cantilever are independent of the ratio between the distributed and localized forces. Currently, the approaches for the calibration of the cantilever spring constant are almost universally based on the determination of the resonant frequencies of the free cantilever 44,45,46,47,48 typically determined using mechanical excitation on the clamped end. From this analysis, we conclude that electrostatic excitation in the non-contact regime can be used to determine resonant frequencies as well, thus providing a convenient approach for cantilever calibration for PFM measurements.…”

Bioinspired lubricating films of cellulose nanofibrils and hyaluronic acid

“…The normal and torsional spring constants of the cantilevers were determined from the analysis of the thermal vibration spectra of the cantilevers before gluing the glass microspheres and the application of Sader's equations. 37,38 Values of about 0.05 N/m and 1.2  10 9 Nm/rad were obtained for the normal and torsional spring constants, respectively. The experiments were carried out in the following media: i) phosphate buffered saline (PBS), pH 7.4 and high ionic strength (10 mM Na 2 HPO 4 , 1.8 mM KH 2 PO 4 , 137 mM NaCl, 2.7 mM KCl); ii) phosphate buffer (PB), pH 7.4 and low ionic strength (10 mM Na 2 HPO 4 , 1.8 mM KH 2 PO 4 ); iii) PBS, pH 3; and iv) PB, pH 3.…”