Static and dynamic calibration of torsional spring constants of cantilevers

Abstract

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.

Quantitative Electromechanical Atomic Force Microscopy

“…Recently, we have used the IDS to independently calibrate the torsional stiffness of cantilevers. 40 Following Jesse et al, 41 we define an "effective" inverse piezosensitivity d eff by…”

“…One thing to note is that the IDS measurements are also capable of resolving the lateral response of the cantilever with some complications as the IDS spot moves away from the symmetry axis of the cantilever, the displacement reflects a sum of the vertical and torsional cantilever motion. Recently, we have used the IDS to independently calibrate the torsional stiffness of cantilevers …”

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

“…where k tip is the spring constant at the tip apex position, k end is the spring constant at the very end of the cantilever, and L tip is the length of the cantilever from the base to the tip. Equation 2is for a beam-shaped cantilever geometry, but it has been shown to be a good approximation for all cantilever geometries [14,38]. When using Equation (1), the cantilever dimensions and Young's modulus are the primary sources of uncertainty.…”

“…The calibration of the torsional spring constant of AFM cantilevers has also been explored and is important to the study of friction and tribology on the nanoscale. Recent work by Cafolla et al involved a method to determine the torsional spring constant using the properties of viscous environments [37], while Labuda et al developed a technique for calibrating the static and dynamic torsional spring constant of AFM cantilevers by directly measuring the thermally driven motion of the cantilever with an interferometer [38]. Thoren et al also demonstrated the calibration of the flexural and torsional spring constants and the deflection sensitivities for AFM cantilevers using a noninvasive thermal noise method [39].…”

Optical direct detection of thermal vibrations of ultralow stiffness micro-nano structures.

“…where ∅ is the angle of reflection. [20,26,75], bead position and stiffness determination of optical traps [76][77][78]. QPDs also have been used to detect position in light detection and ranging (LIDAR), robotics [79], inter-satellite communication [80], and laser space communication [81].…”