Biomedical Engineering Reference
In-Depth Information
Q
(
x
,
y
,
z
,
t
) =
L
21
E
(
x
,
y
,
z
,
t
) −
K
∇
p
(
x
,
y
,
z
,
t
)
,
(6.4)
where
σ
and
K
are the material electric conduc-
tance and the Darcy permeability, respectively.
The cross-coefficient is usually
L
=
L
12
=
L
21
. The
simplicity of the preceding equations provides
a compact view of the underlying principles
of actuation, transduction, and sensing of the
IBMCs, as also shown in
Figures 6.7 and 6.8
.
When the
direct
effect is measured (actuation
mode,
Figure 6.7
), since ideally the electrodes
are impermeable to ion species flux, it is observed
that
Q
=
0
. This gives:
L
K
E
(
x
,
y
,
z
,
t
).
∇
p
(
x
,
y
,
z
,
t
) =
(6.5)
This
∇
p
(
x
,
y
,
z
,
t
)
will, in turn, induce a curvature
κ
proportional to
∇
p
(
x
,
y
,
z
,
t
)
. The relationships
between
κ
and the pressure gradient
∇
p
(
x
,
y
,
z
,
t
)
were fully derived and described by de Gennes
et al.
[46]
. Let us just mention that
κ
=
M
/
YI
,
where
M
is the local induced bending moment
and is a function of the imposed electric field
E
,
Y
is the Young's modulus of the strip that is a
function of the hydration
H
of the IPMC, and
I
is the moment of inertia of the strip. Note that
locally
M
is related to the pressure gradient such
that in a simplified format:
∇
p
(
x
,
y
,
z
,
t
)
=
M
/I
= κ
E
.
(6.6)
Now from Eq.
(6.6)
it is clear that the vectorial
form of curvature
κ
E
is related to the imposed
electric field
E
by:
κ
E
=
(
L
/
KY
)
E
.
(6.7)
FIGURE 6.11
Hydrated cations migrate away from local-
ized anode electrode toward the cathode electrode, causing
the IBMC strip to bend toward the anode electrode.
Based on this simplified model, the tip-bending
deflection
δ
max
of an IPMC strip of length
l
g
should be almost linearly related to the imposed
electric field due to the fact that
water flux). The conjugate forces include the
electric field
E
and the negative of the pressure
gradient
∇
p
. The resulting equations are:
l
g
+
Δ
2
MAX
=
2Δ
MAX
/
l
g
.
Κ
E
=
2Δ
MAX
/
(6.8)
The experimental deformation characteristics
depicted in
Figure 6.6
are clearly consistent with
the predictions obtained by the above linear
J
(
x
,
y
,
z
,
t
) = Σ
E
(
x
,
y
,
z
,
t
) −
L
12
∇
p
(
x
,
y
,
z
,
t
),
(6.3)
