  The dynamical properties of an oscillating tip-cantilever system are now
widely used in the field of scanning force microscopy. The aim of the present
work is to get analytical expressions describing the nonlinear dynamical
properties of the oscillator in noncontact and intermittent contact situations
in the tapping mode. Three situations are investigated: the pure attractive
interaction, the pure repulsive interaction, and a mixing of the two. The
analytical solutions obtained allow general trends to be extracted: the
noncontact and the intermittent contact show a very discriminate variation of
the phase. Therefore the measurement of the phase becomes a simple way to
identify whether or not the tip touches the surface during the oscillating
period. It is also found that the key parameter governing the structure of the
dynamical properties is the product of the quality factor by a reduced
stiffness. In the attractive regime, the reduced stiffness is the ratio of an
attractive effective stiffness and the cantilever one. In the repulsive regime,
the reduced stiffness is the ratio between the contact stiffness and the
cantilever one. The quality factor plays an important role. For large values of
the quality factor; it is predicted that a pure topography can be obtained
whatever the value of the contact stiffness. For a smaller quality factor, the
oscillator becomes more sensitive to change of the local mechanical properties.
As a direct consequence, varying the quality factor, for example with a vacuum
chamber, would be a very interesting way to investigate soft materials either
to access topographic information or nanomechanical properties.
