Experimental APD curves (a) and (b) below Ac and (c) and (d) at Ac where bi-stability is present. (a) and (b) Below Ac the attractive regime prevails throughout both on approach (red) and retraction (blue). This can be deduced by observing that at other than A/A0 ≪ 1 (light blue colored region) the phase always lies above 90°. (c) and (d) As the free amplitude reaches Ac, a discrete transition between force regimes is observed. Note that the phase lies below 90° in some regions of the curve (and at other than A/A0 ≪ 1). At this point two (c) phase and (d) amplitude branches co-exist for a given separation zc. The co-existence of these two branches implies that the system displays bi-stability. Experimental parameters: cantilever model AC240TS, f0 = f ≈ 70 kHz, Q ≈ 150, and k ≈ 2 N/m. For (a) and (b) A0 ≈ 20 nm while for (c) and (d) A0 ≈ 25 nm. The sample is aluminum (Al).
Experimental (dashed grey, continuous black, and dotted black markers) and predicted, by the Rn law (continuous, dashed, and dotted blue lines), values of Ac as a function of tip radius R for a variety of samples. Each marker shows the data obtained with a different cantilever. The dashed grey markers show ranges of Ac obtained for Si, Al, Qu, Mi, and Gr samples (for a given cantilever for each marker); the continuous black markers show ranges of Ac obtained on a Mi sample (one cantilever per marker) and the dotted black markers show ranges of Ac obtained on an Al sample (one cantilever per marker). The predicted Ac curves have been found by fitting the experimental data. Note that the dark (blue) colored region approximately lies in the region of R ≤ 10 nm (Ac ≤ 40 nm) and that the light blue colored region lies approximately in the 10 ≤ R ≤ 20 nm (40 ≤ Ac ≤ 70 nm) region. The experimental parameters are: cantilever model AC240TS, f0 = f ≈ 70 kHz, Q ≈ 150, and k ≈ 2 N/m.
Selection of SEM scans obtained for some of the cantilevers used in Fig. 2 after acquiring the experimental data; AC240TS (Olympus) cantilevers. The tip radius R has been deduced by fitting a sphere at the end of the tip and measuring its radius. A range for R has been allowed however in order to account for errors (shown in the markers in Fig. 2) and the mean value in these measurements has been used to produce the curves of best fit in Fig. 2.
Experimental (continuous black markers) and predicted by the Rn law (continuous blue line) values of Ac as a function of tip radius R for a mica sample. The experimental parameters for the cantilevers used to obtain the data (AC160TS) are: f0 ≈ 300 kHz, Q ≈ 500, and k ≈ 40 N/m.
SEM scans of a selected number of tips obtained after acquiring the experimental data in Fig. 4. The tips shown in (b) and (c) were coated with a 2 nm thick aluminum layer to provide enhanced resolution in the SEM.
Sequence of APD curves obtained on a mica sample with an AC160TS cantilever where the corresponding amplitudes ((a), (c), (e)) and phases ((b), (d), (f)) are shown. The approach and retractions curves are shown in red and blue, respectively. The attractive and the repulsive regimes, and force transitions in the curves can be readily distinguished in both the amplitude and phase curves. These occur at different tip-sample separations during approach and retraction. Furthermore, step-like jumps in both amplitude and phase are observed. These are also a characteristic of bi-stability that can be used to establish if force transitions have occurred. The experimental parameters are: cantilever model AC160TS, f0 = f ≈ 300 kHz, Q ≈ 500, and k ≈ 40 N/m. The sample is mica.
SEM image of the tip of an AC160TS cantilever where large amounts of contamination and/or tip damage are observed. This type of contamination can lead to false predictions of tip radius using the Ac method. Nevertheless, this high level of contamination is not standard and/or typical in AFM experiments, and, in particular, in new cantilevers or tips that have not been submitted to hard interactions.
Summary of the power laws for the Ac and R dependencies for two standard cantilever models: AC240TS and AC160TS (Olympus). The expressions for Ac (2) and R (3) are given for two specific materials, Si and Gr, for the AC240TS cantilever model (k ≈ 2 N/m). The values in the expressions in the table correspond to the curves of best fit in Fig. 2. The errors in the prediction of the tip radius R from Ac are shown in the table. These are the (1) average and (2) maximum errors for R excluding any possible SEM errors and the (3) average and (4) maximum errors including and possible SEM errors. Excluding SEM errors in the table implies assuming that the error in R due to SEM scanning is zero. Thus the only source of error is the experimental range of Ac. Including the SEM errors implies that part of the error in the prediction of R from the expressions in the table is due to inaccuracies in the SEM measurements. Similar expressions accounting for five standard materials, Si, Al, Mi, Qu, and Gr, are also given. Even when accounting for these different materials the errors are relatively small. This exemplifies the relevance of R for a range of cantilever-sample systems.
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