Biomedical Engineering Reference
In-Depth Information
assumption that the ion diffusion is dominant
over the thickness of the membrane with respect
to two other dimensions; so the only spatial
dependence is on
x
(thickness across the mem-
brane). On the other hand, the ion species that
contribute in transport phenomena are cations.
Based on these assumptions, the one-dimen-
sional form of the Nernst-Planck equation for
cation transport will be:
partial differential equation for the kinetics of
charge transport:
∂
t
=
D
∂
2
Ρ
∂Ρ
∂
x
2
−
ΑΡ, 0<
x
<
h
,
(6.18)
where
Α
=
DF
2
RT
Ε
c
−
and
h
is the thickness of the
membrane.
This concludes the formulation of charge
dynamics in chitosan/IPMC artificial muscles.
This formulation indicates that the various
dynamic phenomena in IBMCs are describable
by well-known mathematical models.
∂
c
+
∂
t
∂
c
+
∂
x
+
RT
c
+
∂
V
F
=
D
∂
∂
x
(6.13)
.
∂
x
Because we are interested in the charge density
dynamic of material to relate it to electric cur-
rent, we rewrite Eq.
(6.10)
in terms of
Ρ
using
Eq.
(6.13)
, which gives us the following equation:
6.5 CONCLUSIONS
The properties and characteristics of ionic
biopolymer-metal nanocomposites as bio-
mimetic multifunctional distributed nanoac-
tuators, nanosensors, nanotransducers, and
artificial muscles were discussed. Fundamental
properties of biomimetic distributed nanosens-
ing and nanoactuation of ionic polymer-metal
composites and IBMCs were elaborated on, and
some recent advances in the manufacturing tech-
niques and 3-D fabrication of IBMCs were pre-
sented. Further, two modeling and simulation
methodologies for actuation, sensing, and trans-
duction of IBMcs were described. Procedures
on how biopolymers such as chitosan and per-
fluorinated ionic polymers can be combined
to make a new nanocomposite with actuation,
energy harvesting, and sensing capabilities were
also described. The fundamental properties and
characteristics of biopolymeric muscles as artifi-
cial muscles were presented. Two ionic models
based on linear irreversible thermodynamics
as well as charge dynamics of the underlying
sensing and actuation mechanisms were also
presented.
F
2
RT
∂Ρ
∂
t
=
D
∂
∂Ρ
∂
x
+
Ρ
F
+
c
−
∂
V
∂
x
.
(6.14)
∂
x
Since the anion concentration
c
−
is fixed across
the membrane thickness, we can expand
Eq.
(6.14)
into following form:
∂
2
Ρ
∂
x
2
+
F
2
∂
2
V
∂Ρ
∂
t
=
D
Ρ
∂
x
2
+
F
∂Ρ
∂
x
∂
V
∂
x
F
+
c
−
.
RT
RT
(6.15)
After assuming that the nonlinear terms
∂Ρ
∂
x
and
RT
Ρ
∂
2
V
F
RT
∂
V
∂
x
∂
x
2
are much smaller than the
linear terms, Eq.
(6.15)
becomes:
∂
2
Ρ
∂
x
2
+
RT
c
−
∂
2
V
F
2
∂Ρ
∂
t
=
D
.
(6.16)
∂
x
2
This equation is the linearized form of the
Nernst-Planck equation for ionic polymers.
The Poisson equation is now given by:
∂
2
V
∂
x
2
+
Ρ
Ε
=
0.
(6.17)
Acknowledgment
This work was partially supported by Environmental
Robots Inc.
Substituting the electric field term from
Eq.
(6.17)
into Eq.
(6.16)
, we have the
following Poisson-Nernst-Planck governing
