John E. Sader

Calibration of rectangular atomic force microscope cantilevers

A method to determine the spring constant of a rectangular atomic force microscope cantilever is proposed that relies solely on the measurement of the resonant frequency and quality factor of the cantilever in fluid (typically air), and knowledge of its plan view dimensions. This method gives very good accuracy and improves upon the previous formulation by Sader et al. [Rev. Sci. Instrum. 66, 3789 (1995)] which, unlike the present method, requires knowledge of both the cantilever density and thickness.

Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope

The vibrational characteristics of a cantilever beam are well known to strongly depend on the fluid in which the beam is immersed. In this paper, we present a detailed theoretical analysis of the frequency response of a cantilever beam, that is immersed in a viscous fluid and excited by an arbitrary driving force. Due to its practical importance in application to the atomic force microscope (AFM), we consider in detail the special case of a cantilever beam that is excited by a thermal driving force. This will incorporate the presentation of explicit analytical formulae and numerical results, which will be of value to the users and designers of AFM cantilever beams.

Method for the calibration of atomic force microscope cantilevers

The determination of the spring constants of atomic force microscope (AFM) cantilevers is of fundamental importance to users of the AFM. In this paper, a fast and nondestructive method for the evaluation of the spring constant which relies solely on the determination of the unloaded resonant frequency of the cantilever, a knowledge of its density or mass, and its dimensions is proposed. This is in contrast to the method of Cleveland et al. [Rev. Sci. Instrum. 64, 403 (1993)], which requires the attachment of masses to the cantilever in the determination of the spring constant. A number of factors which can influence the resonant frequency are examined, in particular (i) gold coating, which can result in a dramatic variation in the resonant frequency, for which a theoretical account is presented and (ii) air damping which, it is found, leads to a shift of -4% in the resonant frequency down on its value in a vacuum. Furthermore, the point of load on the cantilever is found to be extremely important, since a small variation in the load point can lead to a dramatic variation in the spring constant. Theoretical results that account for this variation, which, it is believed will be of great practical value to the users of the AFM, are given.

Accurate formulas for interaction force and energy in frequency modulation force spectroscopy

Frequency modulation atomic force microscopy utilizes the change in resonant frequency of a cantilever to detect variations in the interaction force between cantilever tip and sample. While a simple relation exists enabling the frequency shift to be determined for a given force law, the required complementary inverse relation does not exist for arbitrary oscillation amplitudes of the cantilever. In this letter we address this problem and present simple yet accurate formulas that enable the interaction force and energy to be determined directly from the measured frequency shift. These formulas are valid for any oscillation amplitude and interaction force, and are therefore of widespread applicability in frequency modulation dynamic force spectroscopy.

Normal and torsional spring constants of atomic force microscope cantilevers

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 (1993)], and the unloaded resonance technique of Sader et al. [J. E. Sader, J. W. M. Chon, and P. Mulvaney, Rev. Sci. Instrum. 70, 3967 (1999)]. 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 torsion...

Experimental validation of theoretical models for the frequency response of atomic force microscope cantilever beams immersed in fluids

Detailed measurements of the frequency responses of a series of rectangular atomic force microscope (AFM) cantilever beams, immersed in a range of fluids, have been performed to test the validity and accuracy of the recent theoretical model of Sader [J. Appl. Phys. 84, 64 (1998)]. This theoretical model gives the frequency response of a cantilever beam, that is immersed in a viscous fluid and excited by an arbitrary driving force. Very good agreement between experimental measurements and theoretical calculations is found for all fluids considered. Furthermore, a critical assessment of the well-known inviscid model is presented, which demonstrates that this model is not applicable to AFM cantilever beams in general.

Theoretical analysis of the static deflection of plates for atomic force microscope applications

The analysis of the static deflection of cantilever plates is of fundamental importance in application to the atomic force microscope (AFM). In this paper we present a detailed theoretical study of the deflection of such cantilevers. This shall incorporate the presentation of approximate analytical methods applicable in the analysis of arbitrary cantilevers, and a discussion of their limitations and accuracies. Furthermore, we present results of a detailed finite element analysis for a current AFM cantilever, which will be of value to the users of the AFM.

Parallel beam approximation for V-shaped atomic force, microscope cantilevers

Due to its simplicity, the parallel beam approximation (PBA) is commonly used in the analytical evaluation of the spring constant of V‐shaped atomic force microscope (AFM) cantilevers. However, the point of contention regarding the validity of the PBA is as yet an unresolved issue, which has been exacerbated by some recent contradictory reports. In this paper, we present a detailed investigation of the deflection properties of the V‐shaped AFM cantilever, and in so doing, show that the PBA is in fact a valid and accurate approximation, provided the width and length of the parallel rectangular arms are chosen appropriately. As a direct consequence of this finding, we obtain exceedingly simple yet accurate formulas for the V‐shaped cantilever, which will be of value to the users of the AFM.

Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope: Arbitrary mode order

The frequency response of a cantilever beam is well known to depend strongly on the fluid in which it is immersed. In this article, we present a theoretical model for the frequency response of a rectangular cantilever beam immersed in a viscous fluid that enables the flexural and torsional modes of arbitrary order to be calculated. This extends the previous models of Sader and Green [J. Appl. Phys. 84, 64 (1998); 92, 6262 (2002)], which were formulated primarily for the fundamental mode and the next few harmonics, to the general case of arbitrary mode order by accounting for the three-dimensional nature of the flow field around the cantilever beam. Due to its importance in atomic force microscope applications, results for the thermal noise spectrum are presented and the influence of mode order on the frequency response investigated.

Rheological measurements using microcantilevers

The use of microcantilevers in rheological measurements of gases and liquids is demonstrated. Densities and viscosities of both gases and liquids, which can range over several orders of magnitude, are measured simultaneously using a single microcantilever. The microcantilever technique probes only minute volumes of fluid (<1 nL), and enables in situ and rapid rheological measurements. This is in direct contrast to established methods, such as “cone and plate” and Couette rheometry, which are restricted to measurements of liquid viscosity, require large sample volumes, and are incapable of in situ measurements. The proposed technique also overcomes the restrictions of previous measurements that use microcantilevers, which are limited to liquid viscosity only, and require independent measurement of the liquid density. The technique presented here only requires knowledge of the cantilever geometry, its resonant frequency in vacuum, and its linear mass density. A simple yet robust calibration procedure is des...