## 3 Results
Figure 1: Loss tangent AFM study on Sapphire samples: (**a**) height image using the 1 st eigenmode, (**b**) Height image using the 2 nd eigenmode, (**c**) topography line of image (**a**), (**d**) topography line of image (**b**), (**e**) loss tangent image using 1 st eigenmode, (**f**) loss tangent image done by 2 nd eigenmode, and (**g**) histograms of loss tangent. The scale bar is 100 nm. Free first amplitude was 100 nm. Free second amplitude was 20 nm. The setpoint for both of the measurements was 60% to ensure repulsive regime imaging. The first eigenmode frequency was 72 kHz and the second eigenmode frequency was 460 kHz. Although Figures 1 and 2 provided a useful insight in regard to selection of excitation frequency for loss tangent AFM images, questions remain about the effect of the oscillation amplitudes, which have a more important role in defining the tip penetrations on samples. In order to systematically study this effect, a Teflon tape is stretched and mounted on a glass slide. Additionally, we performed four different sets of experiments. For each excitation frequency (i.e., either first or second in this case), two different free oscillation amplitudes are selected so that the product of stiffness and oscillation amplitudes stay constant (i.e., \(\mathrm{k_{1}A_{1}=k_{2}A_{2}}\)). This is done by understanding the equation of motion of the cantilever in normalized form [16, 18, 19]:  
\[\frac{\mathrm{d^{2}z}}{\mathrm{d\mathrm{t}^{2}}}=-\mathrm{z}+\frac{1}{Q} \biggl{[}-\frac{\mathrm{d}z}{\mathrm{d\mathrm{t}}}+\cos(\mathrm{t})\biggr{]}+ \frac{\mathrm{F_{ts}}(\mathrm{z_{fs}})}{\mathrm{kA_{0}}} \tag{5}\]  
where \(\mathrm{A_{0}}\) is the free amplitude, \(\mathrm{z}=\mathrm{z(t)}/\mathrm{A_{0}}\) is the dimensionless tip position with respect to the cantilever base, \(\mathrm{z_{fs}}=\mathrm{z_{fs}}/\mathrm{A_{0}}\) is the dimensionless tip-sample distance (\(\mathrm{z_{fs}=z+z_{eq}}\)) where \(\mathrm{z_{eq}}\) is the position of the cantilever above the sample, \(\mathrm{t}=\omega_{0}\mathrm{t}\) is the dimensionless time, \(\mathrm{k}\) is the cantilever force constant, and \(\mathrm{F_{ts}}\) is the tip-sample interaction force. We made the substitution \(\mathrm{A}\approx\mathrm{A_{0}}=\mathrm{F_{0}}\mathrm{Q}/\mathrm{k}\)[2], where \(\mathrm{F_{0}}\) is the amplitude of the excitation force, and we combined the damping and excitation terms with the factor \(1/\mathrm{Q}\). The last term on the right-hand side represents the tip-sample force interaction that is normalized by the product of \(\mathrm{kA_{0}}\). Therefore, depending on the excitation frequency at which we are shaking the cantilever, its spring constant can change. However, the effect of tip-sample force interactions on the dynamics of the cantilever can be controlled by adjusting the oscillation amplitude. Figure 3 represents a set of studies where the effect of excitation frequency and oscillation amplitude on the measured loss tangent values of a PTFE sample is shown. Figure 2: PS-LDPE loss tangent AFM study: (**a**) height image done by 1 st eigenmode (**b**) height image done by 2 nd eigenmode. (**c**) loss tangent image done by 1 st eigenmode, and (**d**) loss tangent image done by 2 nd eigenmode. Images are 6 \(\mathrm{\SIUnitSymbolMicro m}\) by 6 \(\mathrm{\SIUnitSymbolMicro m}\). Free first amplitude amplitude was 100 nm. Free second amplitude was 20 nm. The setpoint for both of the measurements was 50% to ensure repulsive regime imaging. First eigenmode frequency was 70 kHz and second eigenmode frequency was 458 kHz. Figure 3a represents the topography or height image of the measurements. From the bottom to the top of the image, the imaging conditions have been modified. The lowest portion of the figure represents the measurements using first eigenmode frequency and oscillation amplitude of 50 nm. In the second portion of the image, the excitation frequency is kept constant; however, the oscillation amplitude is increased to 100 nm.