Biomedical Engineering Reference
In-Depth Information
in polymers is well modeled by the
following (neo-Hookean constitutive model) equation (Truesdell and Noll, 1965;
Beatty, 1987):
It is well known that the axial stress
σ
(
)
(
)
−
2
σ
=
YC pHT
s
,
,
/
3
λ
−
λ
,
(6.24)
where
Y
is the Young's modulus of elasticity of the ionic gel and is a function of the
concentrations of solvent
C
s
and pH of the gel as well as the absolute temperature
T
.
Assume that, due to the presence of an electrical voltage gradient across the
thickness
t
g
of the gel, the gel strip is bent into a curved strip by a nonuniform
distribution of fixed as well as mobile ions (cations) in the gel. Accordingly, an electric
field gradient may be imposed across the thickness of the gel by charged molecules.
Note from figure 6.11 that the gel acts like an electrical circuit with resident capacitors
and resistors. No inductive properties may be attributed to ionic gels at this stage of
investigation. However, it is well established experimentally (Shahinpoor and Kim,
2000, 2001g, 2002d, 2002h) that an ionic gel possesses a cross-capacitance,
C
g
, and
a cross resistance,
R
g
. Note that the Kirchhoff's law can be written for a gel strip in
the following form:
−
1
VCQRQ Qi
g
=
+
,
↔=
(6.25)
g
where
V
is the voltage across the thickness of the gel, and
Q
is the charge accumulated
in the gel, being the current
i
through the gel strip across its thickness. Equation
(6.25) can be readily solved to yield:
Q
(
)
(
)
QCV
=
1exp.
−
−
tRC
/
,
(6.26)
g
g
g
assuming that, at
t
= 0,
Q
= 0. Equation (6.26) relates the voltage drop across the thick-
ness of the gel to the charge accumulated, which eventually contributes to the
deformation of the gel strip. Thus, equation (6.26) will serve as a basis for the
electrical control of gel deformations. The imposed voltage gradient across the thick-
ness of the gel forces the internal fixed and mobile ions to redistribute, as shown in
figure 6.12.
The deformation characteristics of ionic polymer gels by electric fields have
been theoretically modeled by Shiga and Kurauchi (1990), Shiga (1997), and Doi
et al. (1992). Both formulations relate the change in osmotic pressure to the change
of volume of the gel samples. In their experiments, the gel sample is not chemically
plated but rather placed in an electrolyte solution in the presence of a pair of
cathode/anode electrodes.
In particular, since the deformation of an ionic gel because of the influence of
an imposed electric field is due to redistribution and shift of ions in the ionic gel,
the change in osmotic pressure,
, associated with ion redistribution and concentra-
tion gradients should be considered. The bending deformation of an ionic polymer
Π
