## 1 Introduction  
Since the invention of atomic force microscopy, amplitude modulation (AM-AFM) is known as the most common mode of imaging various types of samples in dynamic mode measurements of AFM. Besides the relative ease of calculating measurements, AM-AFM can provide a wide range of information, such as topography, stiffness, dissipation, and material composition in a single-pass measurement [1, 2, 3]. Unlike contact mode AFM, AM-AFM provides topographical measurement, with a possibility of compositional mapping that includes a lower chance of damaging surfaces [1, 2, 4]. In a conventional AM-AFM measurement, the cantilever is excited at, or near, resonance frequency (i.e., typically the first eigenmode frequency) while the tip-surface distance is modulated by either moving the sample in \(Z\) direction or moving the base of the cantilever to maintain a constant oscillation amplitude. The amplitude signal is used to extract the topographical measurement while the phase signal is used for material characterization purposes [2]. Although AM-AFM can provide some immediate advantages compared to contact mode or other advanced dynamic modes of AFM, it does not provide a direct quantitative measurement of mechanical properties of materials, especially if they are viscoelastic polymers. During the past few decades, there have many studies to show how one can extract quantitative properties of materials from just the amplitude and phase signals of AM-AFM images [5; 6; 7]. Firstly, these attempts require extensive mathematical models to be used to calculate the material properties. Secondly, they require measurements at various oscillation amplitudes, setpoint values, and different tip velocities. Since most of the samples that are of high importance in these measurements are delicate samples, multiple measurements in the same area over the surface might cause permanent damages. Therefore, it remains challenging to obtain reliable and consistent material properties from AM-AFM measurements. A relatively new approach has focused on distinguishing conservative and dissipative tip-sample force interactions in AM-AFM [8; 9]. In doing so, quantities known as virial and dissipated were defined and used in different studies [6; 8; 10]. Although it is still unknown how one can directly relate virial to a mechanical property of a surface, virial is commonly related to the stiffness of tip-sample forces observed by the cantilever while dissipated power is known as energy loss in a full cycle [7]. This method can separate the effect of each category of forces; however, it requires accurate measurements of oscillation amplitude and probe characteristics to be a reliable measurement technique. Additionally, if a higher eigenmode is used to observe compositional contrasts in phase channel, the calibration process becomes more sophisticated. Recently, studies were done that implemented a more general and straightforward method known as loss tangent (\(\tan\delta\)) measurements [11; 12]. The \(\tan\delta\) quantity is a dimensionless value that is processed as an image (i.e., observable) during a conventional AM-AFM mode. This quantity is defined as the ratio of the dissipative and conservative energy contributions during each oscillation cycle of the probe. Although the value does not provide separate quantities for conservative and dissipative energies, the ratio can provide insightful information about the sample under study.