Biomedical Engineering Reference
In-Depth Information
the surface is negligibly distorted from that shown in Figs. 4-7 and 4-9.
The lack of precision in the numerical algorithm used to solve for the
F A ( s ) responses replicates the thermal and other noise present in a
contact experiment and highlights the trade-off in using stiff probes to
investigate the form of tip-surface interaction forces: A stiffer probe
usually provides greater accuracy for the force response but less
precision.
A second, sub-critical condition, k S
k 2 , is usually encountered in
AFM measurements, in which the stability condition k
<−
0 is fulfilled
only for limited domains of the probe position, s , imposed by the
indenter. Within these domains there are stable equilibrium positions for
the tip, z . However, the domains are bounded by instability conditions
that lead to non-equilibrium tip motion between stable equilibrium
configurations. As a consequence, dual equilibrium configurations are
possible within the domains and the configuration of the system depends
not just on the external state conditions but on the path imposed by the
external constraint. A configurational dependence on path rather than
state leads to irreversibility and thus indentation measurements
performed with sub-critical probe springs give rise to adhesive contact
hysteresis. This is shown in Figs. 4-11 and 4-12 , in which the variation in
F ( z ) for s / z 0 = 3.5, s / z 0 = 2.818, s / z 0 = 2.5, and s / z 0 = 1.0 ( Fig. 4-11 ) and
s / z 0 = 1.0, s / z 0 = 2.5, s / z 0 = 2.693, and s / z 0 = 3.5 ( Fig. 4-12 ) for a sub-
critical spring stiffness of k S = 0.7 k 2 .
Figure 4-11 displays the approach segment that leads to a snap-on
instability and adhesion: For s / z 0 = 3.5, the attractive force of the tip-
surface interaction perturbs the tip to a stable equilibrium position
z
< s (the F = 0 position in Fig. 4-10 ) , but the perturbation is not large,
just as in the super-critical case considered in Fig. 4-9 . Also, as before, as
s is decreased the surface attraction becomes greater and the separation
between the imposed probe position and the tip position becomes greater.
However, in this case a critical condition is reached at s / z 0 = 2.818 for
which the tip is at a condition of incipient instability. Thermal
perturbation or any further incremental decrease in s lead to the tip
moving via a non-equilibrium ( F
0 ) path to “snap on” to the surface at
a new stable equilibrium position considerably displaced (towards the
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