3.1. Effect of Acquisition Frequency
Force-indentation curves from low frequency mode FV (20 Hz) and high frequency mode QNM (2 kHz) performed with the sharp probe were plotted and fitted to the Hertz model (
Figure 1). The noise level is much lower in QNM compared to FV. The baseline noise floor is around 20 nN and 80 nN for QNM and FV, respectively. A more striking difference is noticed when curves from QNM and FV on Au are compared (
Figure 1c,f). The noise level in the contact region of FV on Au is so large that it is difficult to define the zero indentation point (
Figure 1c). In contrast, the noise level of QNM on Au is much smaller. In AFM force spectroscopy, the noise is dominated by the thermal noise. The much lower noise level in QNM is presumably due to averaging and possible signal filtering [
8]. While time-dependent mechanical response has been frequently observed on viscoelastic materials such as polymers [
17] and biological materials [
8], further investigation is needed to fully explain the somewhat peculiar behavior of Au. It can be speculated that the observed frequency dependency may be linked to the relatively high plasticity of Au. The high plasticity of gold is evident from the large hysteresis of the force indentation curves from Au (
Figure 1c,f). Exactly how this property has caused the high noise level at low frequency indentation in
Figure 1c is puzzling, as Au is not known to be viscoelastic.
As expected, the lower noise level of QNM compared to FV resulted in a much narrower distribution of Young’s modulus fitted from the data (
Figure 2). While both data follow Gaussian distribution, the spread of FV is much larger than that of QNM, with a standard deviation of 12.3 GPa compared to the 4.6 GPa of QNM. The goodness-of-fit of FV indicated by R
2 is much worse than that of QNM as a result of high noise in the contact region. Assuming the above standard deviation as population values, the sampling numbers required for achieving a 5% error margin at 95% confidence level are 49 and seven for FV and QNM, respectively. This means that reliable measurement can be done with a smaller number of samples for QNM. Overall, also taking into account the higher acquisition frequency, QNM is a much faster technique than FV.
3.2. Effect of Loading Force and Tip Geometry
Loading force or indentation depth dependence is one of the major issues in achieving accurate mechanical measurements with AFM. The low noise level of QNM makes it possible to measure with very small forces, which allows us to investigate the effect of loading force, starting from very small forces. QNM measurements were performed on FS with varying loading force (
Figure 3). When a sharp tip with around 40 nm radius was used, the measured Young’s modulus shows a three-stage evolution with increasing force (
Figure 3b). The Young’s modulus rises sharply as force increases from 0.25 to 0.75 µN in stage one, stabilizes until the force reaches 1.5 µN in stage two, and then rises again until the force reaches 4 µN in stage three. In the later part of stage three (2.5–4 µN), the deviation of data increased. This is visualized on the mechanical map, where the modulus begins to fluctuate, particularly near the edge (
Figure 3a). A possible cause of this fluctuation at the edge is significant plastic deformation, which begins to affect neighboring pixels. The edge pixels do not have neighboring pixels on both sides, so their moduli are different, causing this edge effect (yellow and blue pixels on the lower edge of
Figure 1a). This is also supported by the observation that the approach and retraction part of the force-indentation curves begin to separate at high forces (data not shown). In contrast, when a blunt tip with around 200 nm radius was used, only stage one and two were observed (
Figure 3e), and the onset of stage two is shifted to larger force. As expected, deformation is smaller compared to the sharp tip (
Figure 3c,f), and plastic deformation seems to be absent.
We extended the measurements to HOPG, Si, and Au to check whether the observations on FS were material specific (
Figure 4). Similar trends are observed in all samples with the exception of the sharp tip on Au. For the sharp tip, the measured Young’s modulus rises with increasing loading forces and stabilizes, before entering the plastic range indicated by the expanded error bars of the data. For the blunt tip, the measured Young’s modulus rises and stabilizes with increasing loading forces.
If we assume that plastic deformation can cause overestimation of Young’s modulus at large forces, it requires further explanation that even within the elastic range, i.e., at small forces, the measured modulus increased with loading force (stage one,
Figure 3b,e). Potential causes of this phenomenon could be surface contamination, asperity on the tip apex, or different mechanical properties of materials near the surface. Surface contamination can largely be excluded because similar results were obtained on fresh surfaces and tips. Asperity on the tip apex could have influences on mechanical measurements at small indentations. Local protrusions, for example, can cause overestimation of the tip radius, leading to underestimation of modulus according to Equation (6). To verify this, we measured the height profile of the sharp tip by scanning on a tip characterization sample TGT1 and calculating the area of the contours at different heights (
Figure 5a). The area of the contours at different indentation depths (height from the apex) was then compared with that expected from a sphere (
Figure 5b). While measurement of the tip profile is subject to system instability at this scale, and therefore not perfectly accurate, the comparison suggests that at ultra-small indentations (<2 nm), tip radius tends to be overestimated and therefore Young’s modulus is underestimated. At large indentations, the trend reverses, leading to overestimation of the Young’s modulus. This observation, together with plastic deformation, serves as a plausible explanation for the observed dependency of Young’s modulus on loading force (
Figure 3b), where the modulus rises, stabilizes, and then rises again with increasing loading force. An alternative/additional explanation of the low Young’s modulus at small forces could be surface water. To sum up, the indentation depth dependent behavior of Young’s modulus measurement can be largely explained by a combination of plastic response and non-ideal tip geometry.
Technically, the stabilization of the measured Young’s modulus after the initial increase allows us to determine the loading force for reporting Young’s modulus (
Figure 4, arrows). These plateaus in the modulus vs. loading force curves represent the range where both surface effect and plastic response are negligible. This procedure can be done for both the sharp tip and the blunt tip, although the plateau is narrower for the sharp tip due to early onset of plastic deformation. However, the reported values of Young’s moduli from the sharp tip and the blunt tip are dramatically different (
Table 1). Deviation of the reduced Young’s moduli from reference values ranges from 69% to 122% for the sharp tip, compared with −6% to 13% for the blunt tip (
Table 1). In all cases, the sharp tip significantly overestimates the Young’s modulus, whereas accuracy of the blunt tip is much higher.
The large overestimation of Young’s modulus by the sharp tip even at small forces suggests that significant plastic deformation could have been induced by the high stress associated with a small contact area. Note that the radius of the sharp tip is approximately one fifth of that of the blunt tip, which translates into 25 times as large stress with the same loading force. This hypothesis is confirmed by examining the surface after QNM measurements (
Figure 6). With the exception of HOPG, the sharp tip induced permanent deformation on all other materials (deformation on Au is obscured by surface features, but the right and bottom borders are clear), whereas the blunt tip hardly left any indents after QNM measurements. It has to be noted that the images were recorded with the same tip for QNM, which explains the different lateral resolution of the images from sharp and blunt tips. However, vertical resolution, which addresses the surface deformation, is largely independent of tip size. A similar tip radius effect has been observed for soft materials [
21]. High stress associated with the sharp tip was also believed to have caused the so-called “skin-effect”, i.e., overestimation of Young’s modulus at the surface. Similarly, this effect can be eliminated by using blunt tips [
21].
It is noteworthy to point out that, apart from the factors investigated so far, there are several other factors and assumptions that affect the accuracy of the measurements. Calibration of deflection sensitivity
S could be complicated by indentation in the sapphire, which may not be negligible for stiff cantilevers. We have used reference measurements on FS to calibrate the tip radius. This method has the advantage of eliminating the need for accurate spring constant calibration. However, it is based on the assumptions of the Hertz model: (1) tip apex is spherical; (2) deformation is fully elastic; (3) surface is flat. In reality, it is unlikely that these conditions can be fulfilled at the same time. For example, at small forces, deformation is more likely to be elastic, but defects at the tip apex have a larger influence on the tip shape. Furthermore, tip rotation, a problem specific to AFM, is likely to have an impact. As already mentioned briefly in the introduction, due to geometric constraints of the cantilever-based probe, the tip must rotate and move laterally during indentation. This will cause buckling of the cantilever, which leads to error in force measurement. In addition, the tip might rotate and slide on the surface. This effect is more evident at large cantilever deflections and high sample moduli [
15]. The tip rotation effect may also partially explain the larger error from using the sharp tip compared with the blunt tip. The sharp tip is more likely to dig into the sample and get stuck, whereas the blunt probe could slide on the sample. The latter causes less cantilever buckling. This explanation is speculative however, due to the fact that it is difficult to isolate the tip rotation effect from other factors.
3.3. Imaging Capabilities
The blunt tip holds advantages over the sharp tip for accurate quantitative measurements. However, this comes at the cost of imaging capabilities. We measured TiO
2 nanocrystals with a lateral dimension of 100–200 nm using both the sharp and the blunt tip, and only the sharp tip was able to resolve the morphology of the nanocrystals (
Figure 7). It was also observed that the nanocrystals were more likely to be detached or dislocated by the blunt tip during scanning, probably due to large interaction forces with the blunt tip. The thickness of the nanocrystal is 10.5 nm (
Figure 7a,b), which is sufficiently large compared to the small deformation of 0.9 nm (
Figure 7e, results from the deformation image, which tend to be lower than the actual indentation. Offline analysis from the force curves gives an indentation depth of 1.8 nm, which is still small compared to sample thickness (less than 20%)). This means that the possibility of the substrate effect for mechanical measurements can be excluded. From the mechanical mapping, we can also obtain multidimensional information. The adhesion force from the adhesion image is around 50 nN (
Figure 7c), which lies within the range of the noise floor (
Figure 7f). This indicates that adhesion of the TiO
2 nanocrystal to the tip is low and cannot be accurately measured with this probe. By comparing the Young’s modulus map to the height image, it could be seen that moduli measured on the edge are lower than that on the top due to the inclined edge plane. We should therefore exclude the edge for the calculation of Young’s modulus. The measured Young’s modulus of the TiO
2 nanocrystal from the top plane is 109.4 ± 16.7 GPa, which agrees reasonably well with the 140 GPa measurement for anatase crystalline film measured by IIT [
22].
To achieve accurate elasticity mapping of rigid materials, it is recommended to work in high frequency mechanical mapping mode, starting with small loading forces and increasing the loading force in small steps until the first plateau is reached. Blunt tips are preferred for good accuracy. The trade-off is that their imaging performance and spatial resolution is not as good as sharp tips.