Biomedical Engineering Reference
In-Depth Information
6.5
MICROELECTROMECHANICAL MODELING
OF ASYMMETRIC DEFORMATION OF IONIC GELS
In order to describe electrically induced deformation of ionic polymeric gels with
no water exudation, we consider the process of bending strips of ionic polymeric
gels in a transverse electric field.
Suppose a long, thin, straight strip of an ionic gel of length
l
g
L
, thickness
t
g
, and
width w
g
, is bent into a curve-strip (fig. 6.10) by an imposed voltage gradient across
its thickness
t
g
.
It is assumed that, initially, the gel is in a natural bending-stress-free
state equilibrated at
pH
= 7. Now, referring to figure 6.11, note that the strain
ε
(defined as the ratio of the actual incremental deformation (
l
) to the initial length
(
l
0
) is given by equation (6.19). In fact, equations (6.19) through (6.26) are still valid
in the present approach.
Due to the presence of an electrical voltage gradient across the thickness
t*
of
the gel, the gel strip is bent into a curved strip by a nonuniform distribution of fixed
as well as mobile ions in the gel. Note that an electric field gradient may be imposed
across the thickness of the gel by charged molecules.
In order to mathematically model the nonhomogeneous deformation or bending
forces at work in an ionic polyelectolyte gel strip, a number of simplifying assump-
tions are made. The first assumption is that the polymer segments carrying fixed
charges are cylindrically distributed along a given polymer chain and independent
of the cylindrical angle
δ
(fig. 6.20).
This assumption is not essential but greatly simplifies the analysis. Consider the
field of attraction and repulsion among neighboring rows of fixed or mobile charges
in an ionic gel. Let
r
i
and
Z
i
be the cylindrical polar coordinates with the
i
th row as
an axis such that the origin is at a given polymer segment. Let the spacing in the
i
th row be
b
i
, and let the forces exerted by the atoms be central and of the form
cr
-s
such that the particular cases of
s
= 2,
s
= 7, and
s
= 10 represent, respectively, the
Coulomb, van der Waals, and repulsive forces. Then it can be shown that the
component of the field per unit charge at the point (
r
i
,
Z
i
) perpendicular to the row
is represented in a series such that
θ
2
0
0
.
0
1
∞
r
n
(
)
=
i
RrZ
,,
,
(6.90)
i
i
(
)
(
)
12
s
1
/
+
(
)
2
0
2
Z b
−
+
r
=−∞
i
i
i
This function is periodic in
Z
i
, of period
b
i
, and is symmetrical about the origin.
It may therefore be represented in a Fourier series in the form:
∞
∑
(
)
=
(
)
(
)
(6.91)
RrZ
,
12
/
C
+
C
cos
2
π
mZ b
/
,
i
i
0
i
i
i
i
m
=
1
where
