Biomedical Engineering Reference
In-Depth Information
.
0
0
2
0
0
∞
)
(
α
i
rCos
2
2
π
mZ
/
b
∑
∫
=
(
)
i
i
i
C
2
/
b
dZ
,
(6.92)
mi
i
(
)
(
)
i
1
/
21
s
+
(
)
0
2
Z b
−
+
r
n
=−∞
i
i
i
Evaluating the coefficients
C
mi
,
m
= 0, 1, 2,....
∞
, in equation (6.92), it is found
that, for
m
= 0,
(
)
(
)
r
Γ
12
/
Γ
1
i
C
=
,
(6.93)
oi
b
(
)
i
Γ
32
/
and for
m
> 0,
(
)
)
(
)
)
)
)
=
(
12
/
s
(
(
(
C
4
r
/
b
π
mb r
/
Γ
1 2
/
/
Γ
1 2
/
s
+
1
K
(
)
mi
i
i
i
i i
12
/
s
(6.94)
(
)
(
)
2
π
mr b
/
s
2
π
mZ
/
b
,
ii
i
i i
i
where
and
K
are modified Bessel functions. In order to simplify these expressions,
in the remainder of this section, only the Coulomb types of attraction and repulsion
forces will be considered. With this assumption, the expression for
R
(
r
i
, Z
i
) becomes
Γ
2
π
mr
b
∑
1
(
)
=
(
)
(
)
−
2
RrZ
,
2
r b
/
r
+
4
mK
1
i
Cos
2
π
mZ
/
b
,
(6.95)
i
i
i
i
i
i
i
i
m
=
Now recall that, in an ionic polymer network, many molecular strands are
occasionally cross-linked, oriented, and entangled. Assuming that, in the presence
of an imposed voltage gradient across the thickness of the gel strip the rows of fixed
and mobile ions line up as shown in figure 6.21, the mean field can be obtained by
superimposing the field corresponding to positive charges, namely:
-
-
-
+
-
+
-
-
-
-
-
-
+
+
+
+
+
+
+
+
+
+
+
+
-
-
-
+
+
+
-
-
+
-
-
-
-
-
FIGURE 6.21
Spatial geometry of a local polymer segment with fixed charges.
