Calibration of higher eigenmodes of cantilevers

Abstract

A method is presented for calibrating the higher eigenmodes (resonant modes) of atomic force microscopy cantilevers that can be performed prior to any tip-sample interaction. The method leverages recent efforts in accurately calibrating the first eigenmode by providing the higher-mode stiffness as a ratio to the first mode stiffness. A one-time calibration routine must be performed for every cantilever type to determine a power-law relationship between stiffness and frequency, which is then stored for future use on similar cantilevers. Then, future calibrations only require a measurement of the ratio of resonant frequencies and the stiffness of the first mode. This method is verified through stiffness measurements using three independent approaches: interferometric measurement, AC approach-curve calibration, and finite element analysis simulation. Power-law values for calibrating higher-mode stiffnesses are reported for several cantilever models. Once the higher-mode stiffnesses are known, the amplitude of each mode can also be calibrated from the thermal spectrum by application of the equipartition theorem.

Fast, High Resolution, and Wide Modulus Range Nanomechanical Mapping with Bimodal Tapping Mode

“…A procedure for calibrating the stiffness of the cantilever's first resonant eigenmode as well as higher eigenmodes has been outlined in detail in recent work. 28 The approach involves measurements of the cantilever thermal noise and use of the equipartition theorem. This procedure also allows calibration of the cantilever deflection sensitivity (units nm/V) of both resonant eigenmodes, from which the cantilever oscillation amplitudes in nanometers can be determined.…”

“…The film was known a priori to be composed of four polymers: polyethylene (PE), polypropylene (PP), polyethylene terephthalate (PET) and ethylene vinyl alcohol (EVOH). Calibration of both cantilever resonance spring constants was performed using a procedure outlined elsewhere 28 while the calibration of the tip size was performed on a polystyrene reference sample, where the modulus was assumed to be 3.0 GPa. A notable example of both the sensitivity and range of this technique is shown in Figure 3d where a 50:50 Sn/Pb alloy solder shows a pattern of softer and stiffer regions in the modulus channel that is not visible in topography (topography not shown; see Ref.…”

“…Here, k2 was calibrated by assuming the stiffness-frequency power law relationship of an Euler- 20 Bernoulli beam. 28 Blue pixels represent data points with NaN values, where negative indentation was calculated and was deemed unphysical. Cantilever: Olympus AC40TS with k1 ≈ 0.12 N/m, f1 ≈ 31 kHz, k2 ≈ 10.0 N/m and f2 ≈ 283 kHz.…”

Background data for modulus mapping high-performance polyethylene fiber morphologies

“…3 bottom, where the full amplitude is marked as 500 picometers. (Through proper calibration of the optical lever sensitivity for the second bending mode, this amplitude was found to be 20% larger [5].). An amplitude of this magnitude is a small fraction of the low trigger force curve shown in Fig.…”

“…4, bottom plots) is relevant because it indicates the frequency shift which occurs, the measureable which is proportional to the tip-sample stiffness. According to reference [5], operating with such aggressive setpoints in the first bending mode (50% reduction from the free-air oscillation amplitude) causes the repulsive forces to dominate the second bending mode measurement (frequency shift).…”