Multi-frequency Atomic Force Microscopy based on enhanced internal resonance of an inner-paddled cantilever

In this work, we study the performance of a new cantilever design during multi-frequency tapping mode Atomic Force Microscopy (AFM). The system consists of a base cantilever with an inner paddle which, under harmonic excitation, vibrates like a system of linearly coupled oscillators engaging simultaneously a lower, in-phase and a higher, out-of-phase resonant mode. The cantilever is designed so that the 2nd mode frequency (i.e., the out-of-phase eigenfrequency) coincides with an integer multiple of the fundamental mode frequency, providing the necessary conditions for realization of internal resonance. During tapping mode, the nonlinear tip-sample force activates the internal resonance and thereby amplifies the out-of-phase resonant mode. We study energy transfer from the lower to the higher mode for different cantilever designs in order to optimize the internal resonance and the amplification of the high-frequency component of the measured response. Numerical and experimental examination of the inner-paddled cantilever performance confirms that a 1:2 internal resonance is optimal. AFM scans of cyanobacteria are taken, and the higher mode observables (amplitude and phase) along with the phase of the lower mode reveal differences in branch selection caused by variations in the stiffness that are not detected by the topographical data. In a computational simulation of an AFM scan across a flat sample with stratified stiffness, we show that the amplitude of the higher mode is significantly more sensitive to variations in Young's modulus than the phase of the lower mode. Further, unlike for a conventional cantilever, we show that the phase of the lower mode indeed depends on the sample's Young's modulus in the absence of a dissipative component in the tip-sample interaction force.