Biomedical Engineering Reference
In-Depth Information
Setting
d
k
M
/d
z
= 0
provides the configuration
(
z
2
,
k
2
)
in which the
surface stiffness field has the least value (numerically minimum
stiffness,
k
2
< 0
):
1/(
mn
−
)
m
zz
n
+
2
2
=
(4-26)
2
1
+
n
+
2
F
z
(
mn
++
)(
)
k
=
(4-27)
1
1
2
z
m
+
2
z
1
2
It is interesting to compare this last value with the stiffness of the
surface-tip interaction,
k
0
, at the
z
=
z
0
equilibrium position,
k
0
=−
mn
Φ
0
/
z
2
>
0
;
+
n
2
n
+
1
2
z
k
=−
k
0
(4-28)
+
2
0
mz
2
The quantity
−
k
2
will be seen to be a critical value in determining the
behavior of the system. (As noted above, the Mie potential is often used
to represent interatomic interactions, in which case the characteristic
interatomic bond quantities discussed in
Chapter 2
are identified with the
intrinsic length, stiffness, and force values above:
d
with
z
0
,
k
bond
with
k
0
,
and
F
max, bond
with
F
1
.)
Once again, the probe will be taken to comprise the tip, interacting
with the surface via
Φ
M
, attached to a spring, described by a harmonic
potential, Φ
S
, with indenter-imposed position
s
. In the previous section,
equilibrium of the tip under these combined influences was shown to be
determined by zero net force on the tip, in this case
F
0, and
stability determined by positive net stiffness, in this case
k
=
k
M
+
k
S
> 0.
possibilities for stability behavior can occur. In the first, super-critical
condition,
k
S
=
F
M
+
F
S
=
k
2
and the stability condition
k
≥ 0
is always fulfilled.
That is, for all positions of the probe imposed by the indenter,
s
, there is
an associated single, stable equilibrium position for the tip,
z
∗
. This is
s
/
z
0
= 1.0,
s
/
z
0
= 0.5 for a super-critical spring stiffness of
k
S
≥−
=
1.3
k
2
.
Search WWH ::
Custom Search