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The atomic force microscope (AFM) (Bennig et al., 1986) has become an important nanoscale characterization tool for the development of novel materials and devices. The rapid development of new materials produced by the embedding of nanostructural constituents into matrix materials has placed increasing demands on the development of new nanoscale measurement methods and techniques to assess the microstructure-physical property relationships of such materials. Dynamic implementations of the AFM (known variously as acoustic-atomic force microscopies or A-AFM and scanning probe acoustic microscopies or SPAM) utilize the interaction force between the cantilever tip and the sample surface to extract information about sample material properties. Such properties include sample elastic moduli, adhesion, surface viscoelasticity, embedded particle distributions, and topography. The most commonly used A-AFM modalities include various implementations of amplitude modulation-atomic force microscopy (AM-AFM) (including intermittent contact mode or tapping mode) (Zhong et al., 1993), force modulation microscopy (FMM) (Maivald et al., 1991), atomic force acoustic microscopy (AFAM) (Rabe & Arnold, 1994; Rabe et al., 2002), ultrasonic force microscopy (UFM) (Kolosov & Yamanaka, 1993; Yamanaka et al., 1994), heterodyne force microscopy (HFM) (Cuberes et al., 2000; Shekhawat & Dravid, 2005), resonant difference-frequency atomic force ultrasonic microscopy (RDF-AFUM) (Cantrell et al., 2007) and variations of these techniques (Muthuswami & Geer, 2004; Hurley et al., 2003; Geer et al., 2002; Kolosov et al., 1998; Yaralioglu et al., 2000; Zheng et al., 1006; KopycinskaMuller et al., 2006; Cuberes, 2009). Central to all A-AFM modalities is the AFM. As illustrated in Fig. 1, the basic AFM consists of a scan head, an AFM controller, and an image processor. The scan head consists of a cantilever with a sharp tip, a piezoelement stack attached to the cantilever to control the distance between the cantilever tip and sample surface (separation distance), and a light beam from a laser source that reflects off the cantilever surface to a photo-diode detector used to monitor the motion of the cantilever as the scan head moves over the sample surface. The output from the photo-diode is used in the image processor to generate the micrograph. The AFM output signal is derived from the interaction between the cantilever tip and the sample surface. The interaction produces an interaction force that is highly dependent on the

Nonlinear Dynamics of Cantilever Tip-Sample Surface Interactions in Atomic Force Microscopy