5.2. Operating Amplitude and spring constant for DFM in Liquid.
It is desirable to operate the DFM at as small an amplitude and with as small a set-point change as possible, in order to minimize disturbance of the sample. This may be seen by considering the energy dissipated in the sample on each cycle in the low frequency limit:
where k is the cantilever spring constant, Aothe amplitude
immediately prior to approach to the surface and A(z) the amplitude at some
distance z from a nominal surface plane (see below). In contrast to DFM in vacuum
or air, resonance effects are small, so the microscope is largely sensitive to changes in
amplitude (as opposed top changes in resonance frequency). Its sensitivity is thus limited
by thermal fluctuations and is not significantly enhanced by operating at resonance.
However, stable operation requires an amplitude sufficient to pull the tip out of
attractive interactions (such as adhesion). In addition to the usual requirement that
(this ensures that the cantilever will not snap into contact) we need
(13)
where A- is the amplitude on the down swing (Fig. 6), FM the
maximum magnetic force (for the case of magnetic drive) and 
is the tip-sample maximal interaction force. This second condition guarantees
that a cantilever which becomes stuck to the surface will be pulled off by the combined
action of the bent cantilever and the applied magnetic field. In non-contact operation
tip-sample adhesion can be avoided, but in normal imaging conditions on rough surfaces
(where accidental contact occurs) a safe value of A0 is between about two and five
nm with k around 1N/m.
Figure 6: Definition of amplitudes of oscillation and cantilever displacements as a DFM cantilever approaches a surface (from Lantz et al. (Lantz, Liu et al. 1999) with permission).
Soft cantilevers (k
0.1 N/m) can give a significant residual
signal on contact because the amplitude of higher bending modes (which do not displace the
tip) can become significant. However, they are probably essential for delicate samples
like the surface of a cell (Dirk Badt, personal communication). This residual signal must
be measured and taken into account when choosing the operating set-point. The softer
cantilevers also have a much lower resonant frequency in water and slower scanning speeds
are required.
It is common practice to calibrate the oscillation amplitude by pushing the cantilever far into the surface and assuming that, in hard contact, the amplitude falls one nanometer for every nanometer the tip advances (i.e., dA/dz=1, see Fig. 6 for definition of quantities). This procedure assumes that the bending profile in oscillation and contact are the same. Although this is not true in general (Sader, Larson et al. 1995) Lantz et al. (Lantz, Liu et al. 1999) have found that, for small triangular cantilevers, the decay of the oscillation amplitude follows the deflection trace in contact at low driving frequencies. This is illustrated in Fig. 7a which shows the bending signal when the tip is driven at 1kHz (well below the resonant frequency of 14.8kHz in water). The lower part of the swing lies on the line drawn through the contact part of the signal for which dA/dz=1. This is not the case at higher frequencies. dA/dz for the same cantilever, driven at resonance, is 1.64 (Fig. 7b). The increased gradient comes about from the dissipation of stored energy as is clear from the decay of both the top and bottom portions of the swing. The Q for this cantilever (k=2N/m) was about 3. When a higher Q cantilever (k=20N/m, Q=12 in water) was driven at resonance, the gradient was nearly 2, the limit appropriate for symmetrical decay of the amplitude (Fig. 7c). Thus, amplitude calibration must first be carried out at low frequency in order to avoid these complications.
