Biomedical Engineering Reference
In-Depth Information
∞
∑
2
π
mr
b
(
)
=
(
)
(
)
−
2
i
RrZ
,
2
rb r
/
+
4
mK
cos
2
π
mZ
/
b
,
(6.96)
+
i
i
i
i
i
1
i
i
i
m
=
1
and corresponding to negative charges, namely:
(
)
=
RrZ
,
−
i
i
(6.97)
∞
∑
2
.
π
mr
b
0
0
2
0
0
(
)
(
)
(
)
+
(
)
i
−
2
2
rb r
/
+
4
mK
cos
2
π
mZ
/
b
Z
1 2
/
b
,
i
i
i
1
i
i
i
i
i
m
=
1
The resulting field for a pair of rows is
∞
135
2
π
mr
b
)
=
(
)
(
(
)
2
i
RrZ
,
16
π
/
b
mK
cos
2
π
mZ
/
b
i
,
(6.98)
i
i
i
1
i
i
m
=
,, ...
The total field due to the presence of
N
strands is then given by
N
∞
2
π
mr
b
∑
∑
(
)
=
(
)
i
*
−
2
RZ
η
,
16
π
b
K
cos
2
π
mZ
/
b
],
(6.99)
i
1
i
i
i
i
=
1
m
=
135
, , ,...
where
is the cross-section variable defined before.
For simplicity, let us assume that all
η
α
i
,
Z
i
, and
r
are equal to
α
,
Z
, and
η
,
respectively, where
is defined as the mean interior ionic distance in the gel. With
this assumption, equation (6.99) reduces to
α
∞
135
2
πη
m
b
(
)
=
(
)
]
*
−
2
RZ
η
;
16
π
b
K
cos
2
π
mZ
/
b
,
(6.100)
1
m
=
,, ...
The mean Coulomb's attraction or repulsion force associated with the mean field
R*
(
η
,
Z
) is then given by
F
(
η
,
Z
) such that
(
)
=
(
)
2
*
FZQR Z
η
,
η
,
,
(6.101)
where
Q
corresponds to total charge between a pair of adjacent ionic surfaces in the
gel strip. This force is repulsive (positive) or attractive (negative) according to
whether like or unlike charges lie in the adjacent rows of charges.
Experimental observations (Shahinpoor and Kim, 2001g) on bending of ionic
gels in the presence of a voltage gradient generally indicated no gross motion in the
direction of the field, suggesting that the force field
F
(
,
Z
) along the long axis of
the gel may be nonuniformly distributed. These facts are also suggested in figures
6.22, 6.23, and 6.24. In fact, the nature of
F
(
η
η
,
Z
), namely:
