چکیده

Contact resonance atomic force microscope (CR-AFM) methods are relatively new measurement techniques used to quantify the elastic and viscoelastic properties of numerous materials such as polymers, elastomers, metallic glasses, asphalt, and biological materials. This technique uses vibration models of the probe and tip-sample contact models to determine the sample properties from the frequency behavior. More recently, AFM U-shaped probes have been developed to allow the in-plane and out-of-plane tip-sample motion to be excited independently at the same location, although they have much more complex resonances. To interpret the experimental data of CR-AFM, we have solved the boundary value problem of the system and validated that with the computational simulation using the finite element method (FEM). A simplified analytical model of the U-shaped probe based on two beams clamped at one end and connected with a perpendicular cross beam at the other end is described her. This three-beam model (TBM) for the case in which the tip is not in contact with the sample is first solved, and the results show good agreement with the FEM solution. After that, the sample contact as three orthogonal Kelvin-Voigt elements are added in the model, and the resonant frequencies and peak widths are well agreed with the FEM solution. Finally, the contact resonant peaks acquired from the experiments are analyzed to obtain the stiffness and damping of the in-plane and out-of-plane of polymers, and in particular, the loss tangent of several polymers such as high-density polyethylene and polystyrene are quantified. We expect the results of this study to allow reliable measurements of the viscoelastic properties of materials with spatial resolution on the order of tens of nanometers.