Biomedical Engineering Reference
In-Depth Information
irreversible thermodynamics formulation that is
also consistent with Eqs.
(6.7) and (6.8)
in the
steady-state conditions and has been used to esti-
mate the value of the Onsager coefficient
L
to be
of the order of 10
−
8
m
2
/V s. Here, a low-frequency
electric field is used in order to minimize the
effect of back diffusion of loose water under a
step voltage or a DC electric field. Other param-
eters have been experimentally measured to be
K
∼
10
−
18
m
2
/CP and
σ
∼
1 S /m
[19]
. The next sec-
tion presents another approach to dynamic mod-
eling of IBMCs in the context of charge dynamics
and Poisson-Nernst-Planck equations.
a bending deformation induces ion diffusion
across the membrane that results in a transient
electric current in a short period of time
(milliseconds) and also an electric potential
across the electrodes plated on the IBMC. The
exact mechanism that causes ion diffusion due
to mechanical stimuli has been investigated by
Porfiri
[29]
by considering the charge dynamics
and micromechanics of ion diffusion in ion
channels of a porous Nafion®
®
membrane.
The electric potential between two electrodes
lasts for a few seconds. Again, by using Eq.
(6.3),
the phenomenon can be described as follows:
After moving cations to one side of membrane
and generating an electric signal, a difference in
ion concentration is created that causes ion diffu-
sion at the reverse side, and ions tend to distribute
evenly across the membrane to maintain a more
stable condition. This causes the induced electric
potential to disappear after a few seconds.
6.4.2 Modeling Charge Dynamics and
Actuation/Sensing Mechanisms of
Biopolymers (IBMCs)
As an external voltage is applied at both sides of
the chitosan/IBMC membrane, an electric field
gradient across the membrane is induced. This
is in accordance with the Nernst-Planck equa-
tions
[47, 53, 55]
such that
6.4.3 Poisson-Nernst-Planck Equation
for Charge Dynamics
The most general governing equations for
charge kinetic of ionic polymers are the Poisson-
Nernst-Planck equations
[53-56]
. The equations
are:
ZF
RT
C
∇
V
J
=−
D
∇
C
+
(6.9)
,
where
J
is the flux of ionic species in mol/(m
2
s),
C
is the concentration of ionic species in mol/
m
3
,
V
is the electric potential field in volt,
D
is
the diffusion coefficient in m
2
/s,
Z
is the valence
of ionic species,
F
is the Faraday constant,
R
is
the universal gas constant, and
T
is the absolute
temperature in Kelvin. Note that based on the
Nernst-Planck equation
(6.9)
, the second term on
the right side acts as an external force that causes
the movement of ions and results in changing
ion concentrations across the membrane. The
difference in ion concentrations results in lateral
expansion and contraction of the biopolymer
that consequently creates a mechanical pressure
gradient due to ion diffusion from one side of
the membrane to another side. This results
in bending of the membrane. Now applying
∂
c
∂
t
=
D
∇·
ZF
RT
C
∇
V
(6.10)
∇
C
+
,
∇
2
V
+
Ρ
Ε
= 0,
(6.11)
Ρ
=
F
(
c
+
−
c
−
)
,
(6.12)
where
c
is the general local charge concentra-
tion,
c
+
is the local cation concentration,
c
−
is the
local anion concentration,
ρ
is the local charge
density, and
ε
is the permittivity of the medium.
As generally the thickness of the polymer
membrane is significantly smaller than the other
two lateral dimensions, it is a reasonable
