Biomedical Engineering Reference
In-Depth Information
d
F
M
d
z
n
(
n
1)
A
z
n
+2
+
m
(
m
+
1)
B
.
(4-21)
k
M
=−
=−
+
z
m
+2
The behavior of these force and stiffness fields is also shown in
Fig. 2-1
in
Chapter 2
shows a similarity and a key difference. The
similarity is that the potential, force, and stiffness curves in both cases
are at first sight just similarly-shaped, shifted (to larger
z
) versions of
each other; this similarity of shape is obvious here in the similarity of the
key difference between the
Φ
M
quantities and the
Φ
A
quantities is that
the Mie potential includes an intrinsic length, and thus energy, force, and
stiffness scales. In the previous case, a length scale was only introduced
into the system through interaction between the montonically attractive
Φ
A
and the spring potential,
Φ
S
. Here, interaction between the attractive
(
A
) and repulsive (
B
) terms leads to a intrinsic length scale for the system
in the absence of the spring. Setting
F
M
=
0
provides the equilibrium
surface-tip separation,
z
0
:
1/(
m
−
n
)
mB
nA
.
(4-22)
z
0
=
The potential minimum,
Φ
0
, at
z
=
z
0
is given by
A
z
n
m
−
n
(4-23)
Φ
0
=−
m
Consideration of
Fig. 4-7
provides insight into the physical meaning
of
z
0
, “indentation,” and “separation.” At location
z
=
z
0
the tip is
essentially “on” the surface and
F
M
0
, the surface
repels the tip and a negative applied force would be required to maintain
the tip in equilibrium in this state; the tip is indenting the surface.
For
z
>
z
0
,
F
M
0
. For
z
<
z
0
,
F
M
=
>
0
, the surface attracts the tip and a positive applied force
would be required to maintain the tip in equilibrium in this state; the tip
is separating from the surface. Hence physical “separation” of the tip and
the surface with this form of potential strictly refers to the distance
z
−
z
0
.
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