In most AFM systems, the deflection of the cantilever is measured as a difference between voltages obtained from position-sensitive detector system. To convert the deflection into distance or force units (newtons), the recorded signal (volts) is multiplied by two factors: a photo detector sensitivity (PSD) and cantilever spring constant (fccant). The former converts volts into nanometers, while the latter enables to determine the force value in nanonewtons according to

In reality, to determine the force, the inverse of the PSD (invPSD) is widely applied, since it directly converts nanometers to volts. The proper calibration of the AFM system is crucial for collection of reliable quantitative data. Therefore, both PSD and spring constant should be determined prior to use of the AFM.

Photodetector sensitivity (PSD)

The deflection of the cantilever is recorded using the optical system composed of a laser and a position-sensitive detector. Thus, the deflection is registered as a difference between quadrant voltages and it is expressed in volts. Typically, to calibrate the cantilever deflection, the force curve is recorded at a stiff, non-deformable surface. For such surfaces, after reaching the contact, the deflection directly reflects the position of the sample (represented by a linear sloped curve, Fig. 3.11).

Usually, due to surface repulsion and/or contamination, the fraction (around 10-20%) of the linear part of the approach just after contact is excluded from the linear regression. In many biological applications, where the force spectroscopy mode is involved, the commonly applied cantilevers have the spring constant within the range of 0.01-0.1 N/m and the typical loading force value does not exceed 30 nN [10]. Assuming that the typical silicon nitride cantilever with a paraboloidal shape (tip radius of 20 nm) indents the flat surface with the load force of 30 nN, the resulting indentation depth is merely of 0.2 nm for glass (this estimation is performed assuming the Hertz mechanics of contact) [10-12]. Therefore, in this range, glass surface can be used as a stiff, non-deformable sample.

The approach part of the force curve, recorded on a glass coverslip

Figure 3.11 The approach part of the force curve, recorded on a glass coverslip (a stiff, non-deformable surface). The photodetector sensitivity (PSD, a conversion factor from volts to nanometers), was determined as a slope of the fitted line (reliable fit range marked by arrows).

The invPSD coefficient is determined as an inverse value of the slope obtained from the linear fit to the region where, after the contact with the surface, the deflection directly reflects the position of the sample. The obtained value reflects the properties of the optical system used for the cantilever deflection. However, one should keep in mind that it changes not only from one AFM system to the other but also from one measurement to another. Fortunately, for a particular AFM system, careful cantilever treatment (especially, during liquid exchange), and proper settings of the optical system assures that its value does not change much (Fig. 3.12).

The invPSD values, determined from a Gaussian fit to obtained distributions for the specific cantilever type (MLCT and OTR-4, with nominal spring constants of 0.01 and of 0.02 N/m, respectively), corresponds to 38.5 ± 2.7 and to 77.3 ± 5.7 nm/V The half width of the distribution at half-maximum height (HWHM), used as an error estimate, is below 10% of the central value for 0.01 N/m and 0.02 N/m cantilevers. Much lower and much larger values of the invPSD conversion factor may indicate broken cantilever. Therefore, in such case, the cantilever must be exchanged.

The distribution of the invPSD values, determined for two cantilever types, MLCT and OTR-4, used in two AFM microscopes

Figure 3.12 The distribution of the invPSD values, determined for two cantilever types, MLCT and OTR-4, used in two AFM microscopes (home build one and the Xe120 from Park Systems, respectively). NT is the total number of cantilevers analyzed. Unpublished data of the author (OTR-4 compared with that from [9] with permission).

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