Spring Constant of the Cantilever

In the non-contact mode (Fig. 6b), AFM acquires the topographic images from measurements of attractive forces in close proximity of the surface, as the tip does not touch the sample and the cantilever oscillates close to the sample surface [12]. This mode is difficult to work with in ambient conditions due to the interference of the capillary forces. Very stiff cantilevers are needed so that the attraction does not overcome the spring constant of the cantilever. However, the lack of contact with the sample means that this mode should be the least invasive and hence cause the least disruption. The disadvantage of this method is that the tip may jump into contact with the surface due to attractive forces. 

In certain cases when adsorption is uniform over the entire cantilever, the spring constant of the cantilever can change. From Eq. (12.5) it is clear that the resonance frequency can change with changes in mass as well as changes in spring constant in a competitive fashion ... 

The interaction forces are balanced by the deflection of the cantilever Fc=kAz. The spring constant of the cantilever depends on the lever geometry and the material used to built the cantilever. For example, the spring constant of a massive... 

The repulsive forces result in a vertical deflection of the cantilever away from the surface. For a hard sample, that does not deform, displacement of the cantilever base, AZc, will be equal to the tip deflection, AZr If the spring constant of the cantilever, kc, is known, one can determine the force exerted on the sample as F=kc Zt. With a typical SFM cantilever, which has a spring constant of 0.01 to... 

Fig. 13. a Rheological model for the cantilever response on applying a displacement modulation to a transducer underneath the sample b the solution to the model gives the ratio of the amplitudes of the tip response Zt to the sample excitation Zc as a function of the logo)/Q)0 for different ratios between the sample stiffness kj and the spring constant of the cantilever kc. Reproduced after [122]... 

When the experimentalist set an ambitious objective to evaluate micromechanical properties quantitatively, he will predictably encounter a few fundamental problems. At first, the continuum description which is usually used in contact mechanics might be not applicable for contact areas as small as 1 -10 nm [116,117]. Secondly, since most of the polymers demonstrate a combination of elastic and viscous behaviour, an appropriate model is required to derive the contact area and the stress field upon indentation a viscoelastic and adhesive sample [116,120]. In this case, the duration of the contact and the scanning rate are not unimportant parameters. Moreover, bending of the cantilever results in a complicated motion of the tip including compression, shear and friction effects [131,132]. Third, plastic or inelastic deformation has to be taken into account in data interpretation. Concerning experimental conditions, the most important is to perform a set of calibrations procedures which includes the (x,y,z) calibration of the piezoelectric transducers, the determination of the spring constants of the cantilever, and the evaluation of the tip shape. The experimentalist has to eliminate surface contamination s and be certain about the chemical composition of the tip and the sample. 

Jump-In. Jump-in arises from attractive forces for both rigid and deformable systems alike, caused by a mechanical instability when the gradient of the force-distance curve is equal to the spring constant of the cantilever. The process occurs quickly. 

The spring constant of the cantilever mnst be measnred directly, as the calcnlated values may not be too reliable. The spring constant, spring, for a beam, with Young s modnlns E, loaded on the end with a rectangular cross section is given as... 

The spring constant of the cantilever could be simulated by applying a known force, i7, at the free end of the cantilever. The force should be centered at the end of the cantilever to avoid any tilt. The spring constant, k = 0.0077N/m, is determined by the applied force, F, and the simulated maximum displacement, z = 0.13pm, by using the formula k = F/z. 

Equations (5.25)-(5.28) demonstrate the importance of the cantilever stiffness with respect to the sample hardness. But to access quantitatively the work of adhesion and the elastic properties of the sample we have to know the spring constant accurately, since Pgff equals kZ with the cantilever deflection, or spring extension needed to unstick the tip. Unfortunately, because of the poor accuracy of the spring constant of the cantilever, rather than quantitative experiments, only comparative studies can be carried out. 

With an AFM, the authors scratch the polymer al high load, between 10" and 10 N which requires a spring constant of the cantilever much larger ... 

Fig. 3 Schematic deflection-versus-piezo translator position curve (Az versus Azc). At large separation, no force acts between tip and sample (a). The cantilever is not deflected. When approaching the surface, it was assumed that a repulsive force is acting (b) and the cantilever is bent upwards. At a certain point, the tip often jumps onto the sample. This happens when the gradient of attractive forces, for example, the van der Waals forces, exceeds the spring constant of the cantilever. After the jump-in, the tip is in contact with the sample surface (c). When retracting the tip, often an adhesion force is observed (d) and the tip has to be pulled off the surface. The deflection-versus-piezo translator position curve has to be transformed as described in the text to obtain a force-versus-distance curve (F versus D).

An important property is the spring constant of the cantilever K. The spring constant can in principal be calculated from the material properties and dimensions of the cantilever. For a cantilever with rectangular cross section, it is... 

Figure 7.1 shows the force-distance curve. There is no interaction between the tip and the surface when the tip is far away from the surface (A in Fig. 7.1). When the tip is close to the surface, there is an attractive force between them. At some point, the force gradient becomes larger than the spring constant of the cantilever, so the tip snaps to the surface (B-C).

The school of Bowen [36] used the AFM technique for direct measurement of the adhesion force between a colloidal probe and membrane surfaces. Colloidal probes were prepared by attaching an 11 pm polystyrene sphere with epoxy resin to a V-shaped AFM tipless cantilever (Fig. 7.2). The AFM allowed the measurement of the force between the colloidal probe and a membrane sample as a function of the displacement of the sample, where a piezoelectric crystal varied the sample displacement. A laser beam reflected from the back of the cantilever fell on a split photodiode that detected small changes in the deflection of the cantilever. To convert the deflection to a force, it was necessary to know the spring constant of the cantilever and to define the zero of the force. The spring constant specified by the manufacturer was 0.4 N m The zero of the force was chosen where the deflection was independent of the piezo position (where the coUoidal probe and the membrane surface were far apart.)... 

All surfaces will jump into contact at a certain distance when the derivative of the potential energy curve for the materials exceeds the spring constant of the cantilever beam. The JKR theory predicts the shape of the bodies in contact just outside the contact zone and it also predicts that the surfaces will jump out of contact at a specific non-zero contact area. One result obtainable from the SFA combined with the JKR theory comes from the force measured at the jump out of contact ... 

The AFM force-distance plots were obtained by oscillating the sample up and down and monitoring the response of the AFM cantilever. The sample had to come into contact with the AFM tip and retract from the tip with each oscillation. Under conditions of small cantilever bending, the force exerted on the cantilever is directly related to the extent of bending by F = kAx, where ax is the distance the tip of the cantilever has moved and k is the spring constant of the cantilever. 

The AFM force-distance plots are similar, but not identical to the force-distance profiles obtained by the surface force apparatus. The bending of the cantilever can follow the force exerted on it by the sample as long as the spring constant of the cantilever exceeds the force gradient of the exerted force. Otherwise, the cantilever jumps into contact with the surface in an attractive regime and in the repulsive regime, is merely pushed a distance equal to the distance the sample is mov, thus... 

The abcissa of the plots is already calibrated by the instrument, only the zero separation distance needs to be determined. The ordinate can be easily cdibrated to show force units because in the linear region of the force-distance plot, the slope equals k, the spring constant of the cantilever aF = kAx (see Figure 2a). Zero force is defined by the flat part of the plot. 

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