Cantilever bending

The equipment required for slow strain-rate testing is simply a device that permits a selection of deflection rates whilst being powerful enough to cope with the loads generated. Plain or precracked specimens in tension may be used but if the cross-section of these needs to be large or the loads high for any reason, cantilever bend specimens with the beam deflected at appropriate rates may be used. It is important to appreciate that the same deflection rate does not produce the same response in all systems and that the rate has to be chosen in relation to the particular system studied (see Section 8.1). 

The amplitude drops to zero when the sample is moved from the point of first contact on a distance of the half of the full amplitude of the free-oscillating probe. From this point, a further motion of the sample will cause the cantilever bending upward, similar to what occurred in the contact mode. If the sample motion is reversed the amplitude increases as shown by a dashed curve in Figure 20.2c. 

Microcantilever deflection changes as a function of adsorbate coverage when adsorption is confined to a single side of a cantilever (or when there is differential adsorption on opposite sides of the cantilever). Since we do not know the absolute value of the initial surface stress, we can only measure its variation. A relation can be derived between cantilever bending and changes in surface stress from Stoney s formula and equations that describe cantilever bending [15]. Specifically, a relation can be derived between the radius of curvature of the cantilever beam and the differential surface stress ... 

The radius of curvature, R, of cantilever bending is related to the deflection, z, and the length of the cantilever, L. Using Eq. (12.1), a relationship between the cantilever displacement and the differential surface stress is obtained ... 

Therefore, the deflection of the cantilever is directly proportional to the adsorption-induced differential surface stress. Surface stress has units of N/m or J/m2. Equation (12.3) shows a linear relation between cantilever bending and differential surface stress. 

The minimum detectable signal for cantilever bending depends on the geometry and the material properties of the cantilever. For a silicon nitride cantilever of 200 pm long and 0.5 pm thick, with E = 8.5 x 1010 N/m2 and v = 0.27, a surface stress of 0.2 mJ/m2 will result in a deflection of 1 nm at the end. Because a cantilever s deflection strongly depends on geometry, the surface stress change, which is directly related to molecular adsorption on the cantilever surface, is a more convenient quantity of the reactions for comparison of various measurements. 

The cantilever bending-technique requires a sensitive displacement detection such as a capacitance probe (Klokholm 1976, 1977), optical interferometry (Sontag and Tam 1986), a tunnelling tip (Wandass et al. 1988) or angular detection (e.g. laser beam deflection, Son-tag and Tam 1986 Trippel 1977 Tam and Schroeder 1988 Betz 1997 Sander et al. 1998). 

Fig.1.a Variation of the interaction force between a flat surface and an isolated atom in the near field of the surface. The dot line corresponds to the lateral displacement of the atom by one interatomic distance. The grey line indicates the force profile sensed by the atom as it moves parallel to the surface plane, b Scheme of a SFM probe a sharp tip mounted on a cantilever. The interaction force Fi=Fs+Fd is a sum of many interatomic interactions, where Fs is the surface force and force Fd results from the sample deformation. The interaction force is balanced by force Fc due to the cantilever bending... 

As discussed in Sect. 2.2.2, FMM images can lose the material contrast when the sample stiffness exceeds the stiffness of the cantilever. In addition, the net signal contains friction effects because of the cantilever bending and the sample indentation. Furthermore, in liquid samples, capillary forces dominate the response at low frequencies [ 127]. These drawbacks can be overcome by operating the microscope above the contact resonance frequencies. In the so-called con-tact-mode scanning local-acceleration microscope the cantilever oscillates at very low amplitudes of ca. 0.1 nm which still provides strong enough contrast with respect to the mechanical properties [122]. Since the response of the canti-... 

Atomic force microscopes have been built in many different versions, with at least six different ways of measuring the deflection of the cantilever [36, 37, 40-42], The commercially available AFM systems use the double photo detector system shown in Figure 7.17 and described by Meyer and Amer [44], Here, a lens focuses a laser beam on the end of the cantilever, which reflects the beam onto two photo detectors which measure intensities T and f2. When the cantilever bends towards the surface, detector 2 receives more light and the difference (h — h) becomes larger. If the tip is scanned over the sample by means of the x- and y-components of the piezo crystal, the difference signal (T — h)/(h + h)... 

Fig. 1. Schematic diagram of a cantilever bending due to differential adsorption of analyte molecules.

There exist a number of readout techniques based on optical beam deflection, variation in capacitance, piezoresistance, and piezoelectricity. Piezoelectricity is more suited for a detection method based on resonance frequency than the method based on cantilever bending. The capacitive method is not suitable for liquid-based applications. The piezoresistive readout has many advantages, and it is ideally suited for handheld devices. 

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