5.6. A numerical model of the high amplitude DFM
The cantilever can be modeled as damped harmonic oscillator colliding with a massive surface held in place with a non-linear spring as described by Lantz et al. (Lantz, Liu et al. 1999). The cantilever is described by a spring constant k, mass m and a Q-factor Q0. The surface is described by a mass M and a Q factor Qs. When the ‘surface’ is indented a distance d, the stiffness is taken to be
where the parameters are fitted to measured stiffness data in the region of
liquid-layer compression (Fig. 11). Thus the ‘surface’ corresponds not to the
underlying hard substrate, but rather to the interfacial layer of fluid that is compressed
in the region where S is on the order of 1N/m. When the tip is a distance 
above this nominal surface, the equation of motion is
where Z is measured from the equilibrium position of the tip and for Z+ and
.
For 
: 
.
and 
. Eqn. 19 can be integrated numerically to obtain the tip trajectory when a stable
state is reached after the initial transients (Lantz, Liu et al. 1999). The trajectory was
Fourier transformed to yield the time-averaged amplitude as a function of frequency. With
an appropriate choice of parameters, the response looks remarkably like that of a damped
harmonic oscillator (see Fig. 14). The results are quite sensitive to the choice of
‘surface’ Q factor. If the damping is increased further, the resonance peak
appears to be cut-off when the tip is quite far from the interface. This occurs when the
low-frequency amplitude is too small for the tip to reach the ‘surface’.
However, as amplitude increases on approaching resonance, the tip eventually touches the
‘surface’. When the damping associated with the surface is very large, further
increase of amplitude as resonance is approached more closely is suppressed, giving rise
to a ‘flat-topped’ response. Thus the harmonic-like amplitude frequency response
is an accidental consequence of the particular properties of the interfaces studied. Even
small increases in surface damping (Fig. 14b) produce results that do not agree with
experiment. The increased damping with increased frequency results in a peak response that
moves to lower frequency as the surface is approached. The data are also quite sensitive
to the function used to describe the interfacial stiffness. The curves shown in Fig. 14a
were obtained with a stiffness decay length, l, of about
1nm, in agreement with the data for mica/water in Fig.11. If this parameter is changed
substantially, the amplitude decay no longer matches the experimentally observed rate
(Fig. 14c).
The numerical simulations confirm the speculative interpretation obtained using the (inappropriate) damped harmonic oscillator. (1) For the mica/water interface much of the amplitude decay on approaching the surface is owing to ‘stiffening’ of the interfacial water layer at a distance some nm from the mica surface. (2) Much of the approach is in the non-contact regime. Significant damping of the cantilever amplitude occurs when the interfacial stiffness at the lowest point of the swing becomes equal to the cantilever spring constant.
