Nanomechanical mapping in air or vacuum using multi-harmonic signals in tapping mode atomic force microscopy.

We present a method by which weak multi-harmonic signals acquired during a normal tapping mode (amplitude modulated) AFM scan of a sample in air or vacuum with standard microcantilevers can be used to map quantitatively the local mechanical properties of the sample such as elastic modulus, adhesion, and indentation. The approach is based on the observation that during the tapping mode operation in air or vacuum, the 0th and 2nd harmonic signals of microcantilever vibration are encountered under most operating conditions and can be mapped with sucient signal to noise ratio. By measuring the amplitude and phase of the driven harmonic as well as the 0th and 2nd harmonic observables, we nd exact analytical formulas that relate these multi-harmonic observables to local mechanical properties for simple tip-sample interaction models. Least squares estimation of the local mechanical properties is performed pixel by pixel. The method is validated through computations as well as experimental data acquired on a polymer blend made of Polystyrene and Polyolen elastomer.