Biomedical Engineering Reference
In-Depth Information
where
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
Γ
and
K
R
(
r
,
Zi
) becomes:
i
2
π
mr
b
∑
−
2
i
(5.8)
Rr Z
(, ) (
=
2
r b
/ )
r
+
4
mK
cos(
2
π
mZ
/)
b
i
i
i
i
i
1
i
i
i
m
=
1
Now recall that in an ionic polymer network, many molecular strands are gen-
erally cross-linked, oriented, and entangled. Assuming that in the presence of an
imposed voltage gradient across the thickness of the gel cylindrical sample, the rows
of fixed and mobile ions line up, the mean field can be obtained by superimposing
the field corresponding to positive charges, namely:
∞
∑
2
π
mr
b
RrZ
(, )
=
2
r b r
/ )
−
2
+
4
mK
i
cos(
2
π
mZ
/)
b
(5.9)
+
i
i
i
i
i
1
i
i
i
m
=
1
and corresponding to negative charges, namely:
RrZ
(, ) (
=
2
rb
/ )
−
i
i
i
i
(5.10)
∞
∑
2
π
mr
b
(
)
(
)
i
−
2
r
+
4
mK
cos
2
π
mZ
/
b
Z
+
( /
1 2
)
b
i
1
i
i
i
i
i
m
=
1
The resulting field for a pair of rows is:
∞
135
2
π
mr
b
(
)
2
Rr Z
(, ) (
=
16
π
/
b
)
mK
i
cos
2
π
mZ
/
b
i
(5.11)
i
i
i
1
i
i
m
=
,, .
The total field due to the presence of
N
strands per unit length of the sample is
then given by:
N
∞
2
π
mr
b
∑
∑
(
)
*
(, )
−
2
i
RrZ
=
16
π
b
mK
cos
2
π
mZ
/
b
(5.12)
i
1
i
i
i
i
=
1
m
=
135
,, .
For simplicity, let us assume that all
b
,
Z
, and
r
are equal to
b
,
Z
, and
r
,
i
i
i
respectively, where
b
is defined as the mean interior ionic distance in the gel so that,
on the average,
N
= 2/
b
. With this assumption, equation (5.12) reduces to:
∞
∑
2
π
mr
b
(
)
*
−
3
RrZ
(, )
=
32
π
b
mK
cos
2
π
mZ
/
b
(5.13)
1
m
=
1
,
22, ...
