5.5. Amplitude decay over a wide frequency range

The situation for high amplitude oscillation over a wide frequency range is more complicated. It was investigated for a silicon cantilever and a mica substrate under water by Lantz et al. (Lantz, Liu et al. 1999). A small white noise signal was added to a larger sinusoidal (5nm p/p) driving signal, the cantilever held at a fixed distance from the surface by feeding back on the sinusoidal signal and the displacement signal was acquired as a function of time. The Fourier transform of these signals (Fig. 12) gave a sharp peak at the driving signal (off scale in Fig. 12) while the white noise drive sampled the response of the cantilever over a broad range of frequencies. These are shown fitted with the damped harmonic oscillator response given by Eqn. 15. The fits suggest that, for the three distances further from the surface (nominally "far", 4.8nm and 3.2nm) the main effect on oscillation amplitude is owing to the stiffening of the medium. Stiffening accounts for the reduction in amplitude at low frequency (i.e., the data of Fig.11 as deduced using the approach curves and Eqn. 18). The same stiffening also accounts for the observed increase in resonant frequency. Agreement with Eqn. 15 is somewhat surprising because the motion is highly anharmonic, the stiffness changing substantially over a cycle of cantilever cycle. Yet the form of the curves is in agreement with data taken at low amplitude as a tip approaches a mica surface (O'Shea, Lantz et al. 1998). We discuss this further with the use of a numerical model in the next section. Close to the surface ("1.6nm" in Fig. 12) damping also increases substantially. The dominance of stiffening (as opposed to damping) may be an unusual property of water. In an organic solvents (OMCTS), increased damping contributes most to reduction of amplitude at higher frequencies, as shown in Fig. 13. This displays the response of the cantilever with a low amplitude drive (0.1 to 0.5nm) with added white noise as a graphite surface is approached (O'Shea, Lantz et al. 1998). Significant reduction of amplitude is first observed at about 3nm from the surface, but Fig. 13 shows that the cantilever resonant frequency is little affected over the entire approach. Thus, in this case the dominant mechanism is enhanced dissipation as the surface is approached. Note that the effect decays on a molecular length scale in contrast to the hydrodynamic damping described by Eqn. 3.

Figure 14: (a) Anharmonic modeling of the dynamic response for approach using interfacial stiffness measured for mica (Fig. 11). Changing the surface damping (b) or the stiffness model (c) yields curves which do not match the experimental results shown in Fig. 12. (From Lantz et al. (Lantz, Liu et al. 1999) with permission.)