The ability to simultaneously map variations in topography and composition (local stiffness, adhesion, charge, hydrophillicity/phobicity, viscoelasticity) of samples in ambient and liquid environments has made dynamic atomic force microscopy (dAFM) a powerful tool for nanoscale metrology. In ambient and vacuum environments, quality factors (Q-factors) of the fundamental resonance are typically large, and the contrast channels in dAFM are relatively well understood. In liquid environments, however, Q-factors are typically low due to cantilever interactions with the surrounding viscous liquid, which introduces a new class of nonlinear dynamics that is accompanied by new contrast channels, such as, higher harmonic amplitudes and phases. In particular, we find that the interpretation of the traditional contrast channels is quite different in low-Q environments compared to high-Q environments. We present a theoretical investigation of the contrast channels in dAFM in the context of frequency modulation and tapping mode dAFM with an emphasis on low-Q environments.
An analysis on the controllability and stability to some fractional delay dynamical systems on time scales with impulsive effects