Cantilever torsion

Figure 5 shows the laser beam path reflected by the torsional cantilever. The incident angle is y on the cantilever surface before the cantilever torsion. When the torsional angle of the cantilever is 6i, reflection ray turned an angle a. Their relationship can be expressed by... 

For lateral force measurements, the constant of the cantilever torsion can be calculated from k=GJ/l, where G is the shear modulus and J is the polar moment of inertia ... 

Fig. 2.31 Schematic illustration of cantilever torsion while (a) sliding up and (b) sliding down on a sloped surface (in the x direction). While sliding across a sloped surface with angle 6, the acting forces (the applied load L, the horizontal tractive force T, the adhesion force A, the reaction force from the surface acting on the tip with a component N in the surface normal direction and a component/(friction force) parallel to the surface) and the torsion momentum M are in equilibrium and depend on the direction of motion - uphill and downhill, denoted here with subscripts u and d, respectively, cp represents the torsion angle of the cantilever, which is proportional to the friction force h and t stand for tip height and cantilever thickness, respectively (reproduced with permission from [18]. Copyright 2006 American Chemical Society)...

M. Reinstadtler, U. Rabe, V. Scherer, U. Hartmann, A. Goldade, B. Bhushan, and W. Arnold, On the nanoscale measurement of friction using atomic force microscope cantilever torsional resonances, Appl. Phys. Lett. 82, 2604 2606 (2003). 

Fig.3. A series of lateral force images. Dark means more torsion, that is, higher fiiction. For each image, the scan direction was from left to right, a), b), and c) show the same flower-shaped domain in different orientations after rotation of the sample underneath the cantilever. In d), the cantilever torsion is plotted as a function of position along a line through the bottom half of the flower in c) for both the left-to-right and right-to-left scans. The use of A and B , indicated in d), is explained in the text.

We observed the effect presented in Fig.3. A Langmuir-Blodgett monolayer deposited on mica forms flower-shaped domains. The molecules within a flower are in the condensed state. The matrix surrounding the flowers is a disordered phase. The condensed domains are comprised of subdomains. Upon rotation underneath the cantilever, the contrast of the subdomains changes in a systematic way. The cantilever torsions marked A and B in Fig.3d were added to ascertain the total friction A+B and subtracted for the friction difference A-B. Upon 360 rotation of the sample, the total friction behaved with C2 symmetry, whereas Ci symmetry was discerned for the friction difference A-B. We denote the C2 behavior by the term anisotropy and the Ci behavior by asymmetry . The anisotropy provides most of the contrast between the subdomains (petals). The small contribution from the asymmetry can be seen in the horizontal petals of Fig.3c. There, the petals have similar orientations with respect to the scan direction, yet exhibit slightly different contrast. 

Tribological Experiments. For tribological measurements, the lateral force lying in the plan of contact between the tip and the surface produces the additional vertical deflection (Figure 2b) or cantilever torsion. Lateral forces arise not only from friction but also from the local surface slope (31, 51). If the sample surface is not flat, the surface normal force will have a component directed laterally and will result in contrast in lateral image. These complications can be avoided by using flat samples whenever possible. 

As AFM has matured, a number of more sophisticated imaging modes have been developed. In lateral force mode (Fig. 3B) the sample is scanned sideways relative to the long axis of the cantilever. Torsional forces exerted on the tip cause the cantilever to twist and consequently to deflect the optical beam horizontally on the photodetector. Recording the lateral deflection or twisting of the cantilever as a function of X-Y position gives a lateral force map. Lateral force images are particularly sensitive to friction force between the tip and sample, and therefore these images are also called friction force maps. 

Figure Bl.19.23. Principle of simultaneous measurement of nomial and lateral (torsional) forces. The intensity difference of the upper and lower segments of the photodiode is proportional to the z-bending of the cantilever. The intensity difference between the right and left segments is proportional to the torsion, t, of the force sensor. (Taken from [110], figure 2.)...

Figure Bl.19.24. Friction loop and topography on a heterogeneous stepped surface. Terraces (2) and (3) are composed of different materials. In regions (1) and (4), the cantilever sticks to the sample surface because of static friction The sliding friction is tj on part (2) and on part 3. In a torsional force image, the contrast difference is caused by the relative sliding friction, Morphological effects may be...

A very practical way to infer the contact area was later developed by Carpick et al. [65] and Lantz et al. [66]. In these experiments, a small (up to nanometer) lateral modulation, djc, is applied to the sample, and torsion of the cantilever is monitored with a lock-in amplifier to detect the lateral force response, dF (Fig. 5). In this way, the lateral stiffness, [51], given by... 

The shape of AFM cantilevers (much thinner than the width) results in torsional deflection when forces push the tip laterally as in friction measurements (when the tip is sliding) or lateral stiffness measurements (when the tip is stuck). 

Fig. 5 —The relation between the rotated angle of reflection beam (a) and the torsional angle of cantilever ( /), (1-rotated angle of the reflection beam, a 2-incident angle, y 3-torsional angle of the cantilever, / 4-reflection surface before torsion of the cantilever 5-reflection surface after torsion of the cantilever).

Through measurement of the lateral-voltage variation of the position detectors, the torsional angle Oj of the cantilever can be given by... 

Then, the corresponding relationship between lateral voltage of the four-quadrant position detectors (F/) and the torsional angle of the cantilever ( di). Thus we can obtain the variation of torsional angle 0i through reading the variation of lateral voltage (F ) from the front panel. 

In 1987 Mate et al. [468] used, for the first time, an atomic force microscope (AFM) to measure friction forces on the nanometer scale (review Ref. [469]). This technique became known as friction force microscopy (FFM) or lateral force microscopy (LFM). To measure friction forces with the AFM, the fast scan direction of the sample is chosen perpendicular to the direction of the cantilever. Friction between the tip and the sample causes the flexible cantilever to twist (Fig. 11.7). This torsion of the cantilever is measured by using a reflected beam of light and a position-sensitive detector in the form of a quadrant arrangement of photodiodes. This new method made it possible for the first time to study friction and lubrication on the nanometer scale. 

Inherent in the oscillation of any object is the existence of different modes of oscillations and of higher harmonics. In addition to the lateral modes there is also the torsional mode. All modes can be detected in the x-y-directions by the PSD (Fig. 4.22). In the case of resonant cantilevers, the different modes have different mass sensitivity (Kim et al., 2001). They are well separated on the amplitude spectrum (Fig. 4.24), and show different sensitivities to absorption of vapors into the selective coating applied to the cantilever (Fig. 4.25). 

Fig. 4.24 Root-mean-square (RMS) amplitude spectrum of cantilever exhibiting lateral and torsional vibration modes and higher harmonics (adapted from Kim et al., 2001)...

Fig. 5. a Schematic representation of the SFM set-up using the optical beam deflection method, b When the tip interacts with the sample surface, the cantilever exhibits deflection perpendicular to the surface as well as torsion parallel to the surface plane. The normal force Fn and the lateral force FL corresponds to the force components which cause the deflection and torsion, respectively. Both responses are monitored simultaneously by the laser beam which is focused on the back side of the cantilever and reflected into a four-quadrant position sensitive detector (PSD)... 

Fig. 9. Torsion of the cantilever, i.e. the lateral force signal FL, upon scanning from left to right a corrugated surface a with a sticky region b...

The lateral force microscope (LFM) is a modification of the standard contact mode SFM [87-90]. In addition to the normal forces, the friction forces exerted on the probe are measured via torsion of the cantilever (Fig. 5). This mode is sometimes called friction force microscopy . LFM can be used in combination with topographic imaging as it shows changes in material as well as enhanced contrast on sharp edges (Fig. 9). In addition to morphology, it provides information on the friction and wear properties (Sect. 3.4). 

Calculation of the critical lateral force is more complicated because traditional LFM does not provide us with easy method to translate current units into force ones. There is no way to define the factor of proportionality until calibration algorithm was developed recently by ourselves [8], The required coefficient depends on design of a microscope, adjustment of the optical system, torsion force constant kL of cantilever and tip height lnp. 

To determine torsion force constant and height of tip we didn t refer to cantilever passport parameters that can be significant varied. The value of kL we found used new the most precise method based on analyzing of amplitude-phase characteristic [to be published] but tip height was determined with the help of optical microscope. We got kL = 77.1 N/m ltip = 10 pm. 

Fig. 2 Schematic representation of the basic detection elements of the scanning force microscope and of the piezoelectric transducers generating the displacement modulations for purposes of dynamic mechanical measurements. The dynamic components of the tip-sample forces resulting from the normal/lateral displacement modulations are detected via the torsion/bending of the microscopic cantilever and the deflection of the laser beam reflected off the rear side of the cantilever. The positional shift of the latter is registered by means of a segmented photo-diode...

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