Biomedical Engineering Reference
In-Depth Information
transport
(with a flux, , we can assume that this term is
water flux). The conjugate
forces include the electric field,
Q
E
, and the pressure gradient,
−∇
p
. The resulting
equation has the concise form of
Jxyzt
(,,,)
=
σ
Exyzt
(,,,)
−
L
∇
pxyzt
(,,,)
(6.131)
12
Qxyzt
(,, )
=
L Exyzt
(,,,)
−
K pxyzt
∇
(,,,)
(6.132)
21
and
K
are the material electric conductance and the Darcy permeability,
respectively. A cross-coefficient is usually
L
=
L
12
=
L
21
. The simplicity of the
preceding equations provides a compact view of the underlying principles of actu-
ation, transduction, and sensing of the IPMNCs, as also shown in figure 6.28.
When we measure the
direct
effect (actuation mode, fig. 6.29), we work (ideally)
with electrodes impermeable to water, and thus we have
where
σ
Q
= 0. This gives:
L
K
∇
pxyzt
(,,,)
=
E xyzt
(,,,)
(6.133)
This will, in turn, induce a curvature proportional to .
The relationships between the curvature and pressure gradient are
fully derived and described in de Gennes et al. (2000). Let us just mention that (1/
∇
pxyzt
(,,,)
κ
∇
pxyzt
(,,,)
∇
pxyzt
(,,,)
κ
ρ
c
)
=
M
(
E
)/YI, where
M
(
E
) is the locally induced bending moment and is a function
-
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
Side chain
Water
Fixed anion
Mobile cation
Hydrated cation-
Na(H
2
O)4
+
FIGURE 6.28
Schematics of the electro-osmotic migration of hydrated counter-ions within
the IPMNC network.
