5.4. Harmonic and Anharmonic analysis of low frequency data
O’Shea et al. (O'Shea, Welland et al. 1994; O'Shea, Lantz et al. 1998) have analyzed data for low amplitude operation using a damped harmonic oscillator model of the cantilever motion in which the displacement amplitude A(z,w), is given by
where FM is the magnitude of the applied
magnetic force, k the cantilever spring constant, 
> an effective interface stiffness, 
an
effective cantilever mass and 
the mechanical Q factor.
Bars over the quantities S, m* and Q indicate
that they are averaged over the motion of the cantilever, which is accurate for small
amplitudes. This approximation breaks down at high amplitude. For example, the low
frequency limit of (15) yields the well known result 
(Pethica
and Oliver 1987), implying that the fractional change in amplitude is constant at a
given z. This is obviously not true at high amplitude where all of the motion that
lies beyond the range of surface forces is unaffected by the approach.
Figure 11: Interfacial stiffness deduced from the approach curves shown in Figs 7a (mica) and 10b (gold) using equation 18. The straight line is the fit used for numerical simulation of the operation of the DFM at high amplitude. (From Lantz et al. (Lantz, Liu et al. 1999) with permission.)
It is straightforward to generalize the result of Pethica and Oliver (Pethica and Oliver
1987) to the high amplitude regime. Consider a cantilever initially at a height h
above a surface (Figure 6) with a maximum upward swing A+ =FM/k and a
maximum downward swing A1-. Moved a distance dz towards the surface (Fig.
6), the new downward swing is A2-. With dA=A1- - A2- and z’=z-(dz-dA)
the force equilibration on the downward stroke is given by
(16)
where we have assumed a symmetric magnetic drive so that the magnitude of the magnetic force on the downward swing is equal to that on the upward swing. If we define a differential force ‘constant’, S(z) by
(17)
then equation 16 yields
. (18)
with repulsive forces defined as negative. Thus, the inverse of the derivative of the approach curve taken at low frequency may be used to estimate S(z). The approach curves of Fig. 7a and 10b were used to generate the values for S(z) displayed in Fig. 11 as described by Lantz et al. (Lantz, Liu et al. 1999). Far from the surface, the stiffness is zero as expected, while close to it, the stiffness rises to the 100N/m values typical of compression of a solid surface (Pethica and Oliver 1987; Sarid 1992). There is, however, a remarkably large region in which the stiffness in on the order of 1N/m, characteristic of compression of a fluid layer (Dhinojwala and Granick 1996; Han and Lindsay 1998; O'Shea and Welland 1998). The data taken over mica show an extended region of exponential decay, indicated by the straight line.
Figure 12: Dynamic response of cantilever at various distances from the surface (a, b, and c on approach curve in inset show set-points for lower three curves). Data taken in water on mica. Solid lines are fits to equation 15. (From Lantz et al. (Lantz, Liu et al. 1999) with permission.)
Figure 13: Low amplitude response curves for the tip approaching graphite under OMCTS. Damping appears to be the dominant interaction in this case. Noticeable damping begins with the tip about 3nm from the surface. (From O’Shea et al. (O'Shea, Lantz et al. 1998) with permission.)
