## 3 Results  
We investigated the effect of excitation frequency on the tip penetration through different surfaces in the context of loss tangent AFM measurement. It is known that higher excitation frequencies can cause higher tip penetration through surfaces [16; 17]. As mentioned, the recent studies in loss tangent AFM field have shown that the higher tip radius can cause higher dissipation, hence, higher loss tangent values. However, these studies have not considered the effect of tip penetration on the surface and the consequence of **effective** tip radius on these measurements. In this work, we examined different types of samples in a systematic way in order to experimentally find the effect of these parameters on loss tangent AFM measurements. Figure 1 represents two sets of experiments using AM-AFM mode, where the amplitude and phase signals are used to generate the loss tangent images. The first round of measurements, shown in the left column of Figure 1, was conducted by exciting the cantilever in AM-AFM with its first eigenmode frequency. In contrast, the second round of measurements, shown in the right column of Figure 1, was performed by exciting the cantilever in AM-AFM with its second eigenmode frequency. The sample understudy in Figure 1 is Sapphire, which is known as a flat and stiff surface. Therefore, one can assume there is no tip penetration in these studies due to surface softness. Figure 1a,b represent topography (i.e., height) images of the same area of the sample conducted with 1 st and 2 nd eigenmode, respectively. Since the sample is relatively flat, scan lines are shown in Figure 1c,d in order to quantitatively compare the differences. Besides the fact that Figure 1d has a larger number of peaks in the scan line (which can be due to imaging conditions), Figure 1c,d show relatively similar topographies as expected. Figure 1e and f show the phase images of the same measurements explained in Figure 1a,b. Although qualitatively they seem similar, the phase images are used to generate loss tangent measurements, which are represented by a histogram in Figure 1g. In this study, it is clear that when we were imaging with the first eigenmode frequency, higher values of \(\tan\delta\) were observed. Additionally, there appear to be two peaks of loss tangent values shown in green histograms of Figure 1g when imaged by the first eigenmode. This observation is linked to the thin layer of humidity that can exist during measurements. Although there are no tip penetrations on this sample, due to higher stiffness of the second eigenmode, the thin layer of humidity can be compressed (or not observed) by the tip-sample force interactions. Consequently, a single peak of loss tangent values is observable for the second eigenmode, which is less dissipative as well. In order to study the effect of tip penetration, we conducted a similar study on a co-polymer sample system (PS-LDPE). PS has an elastic modulus around 2 GPa while LDPE is known to be a softer polymer with 0.1 GPa. This sample allows us to know if there is any difference with the way polymers respond to loss tangent AFM imaging with the first or second eigenmode frequencies. Figure 2a,b show the same area on the sample that used the 1 st eigenmode and 2 nd eigenmode excitation frequencies, respectively. Figure 2c,d represent the loss tangent AFM image done by 1 st and 2 nd eigenmode excitation frequencies, respectively. It is clear from these images that when we excite the cantilever with higher frequency, we see less dissipation compared to when we used the first eigenmode frequency. Although the second eigenmode frequency has a higher stiffness, the tip penetration is lower. Therefore, the effective tip radius results in less dissipation. It should be mentioned that this finding is consistent with the recent findings presented in [PERSON] et al. 's work as well related to the tip penetration by higher eigenmodes [16].