Biomedical Engineering Reference
In-Depth Information
where
φ
(
x
,
y
,
z
,
t
) is an electric potential,
ρ
*(
x
,
y
,
z
,
t
) is an electric charge density, and
*(
x
,
y
,
z
,
t
) is determined by the equiv-
alent weight of the precursor ion-containing polymer and, in particular, the molar
concentration of cations (lithium) and the charge groups in the polymer (sulfonic or
carboxylic), such that
ε
is the dielectric constant of the composite.
ρ
ρ
*( , , , )
xyzt
=
(
ρ
( ,
xyzt
, , )
−
ρ
( ,
xyzt Ne
, , ))
(6.115)
M+
so
3
where
ρ
M+
(
x
,
y
,
z
,
t
) and
ρ
so
3
(
x
,
y
,
z
,
t
) are, respectively, the molal density of cations and
sulfons
N
is the Avogadro's number (6.023
×
10
26
molecules per kilogram-mole in meter-
kilogram-second units)
e
is the elementary charge of an electron (-1.602
×
10
-19
C)
The electric field within the ionic polymeric structure is
Exyzt
(,,,)
=−∇φ
(,,,)
xyzt
(6.116)
Balance of forces on individual cations hydrated with
n
molecules of water inside
the molecular network, clusters, and channels based on the diffusion-drift model of
ionic media due to Nernst and Plank (Nernst-Plank equation) can be stated as:
dv
dt
x
Ne
ρ
E
(,,,)
x y z t
=
N
(
ρ
M
+
n
ρ
M
)(
)
+
N
ρ
η
v
+
Mx
+
M M
+
+
ww
M
+
x
(6.117)
∂
ln(
ρ
+
n
ρ
)
+
∂
∂
P
x
M
+
w
N T
ρ
M
+
∂
x
dv
dt
y
Ne
ρ
E
(,,,)
x y z t
=
N
(
ρ
M
+
n
ρ
M
)(
)
+
N
ρ
η
v
+
My
+
M M
+
+
ww
M
+
y
(6.118)
+
∂
ln(
ρ
+
n
ρ
)
∂
∂
P
y
M
+
w
N T
ρ
M
+
∂
y
dv
dt
z
Ne
ρ
E
(,,,)
x y z t
=
N
(
ρ
M
+
n
ρ
M
)(
)
+
N
ρ
η
v
+
Mz
+
M M
+
+
ww
M
+
z
(6.119)
ln(
)
∂
ρ
+
n
ρ
+
∂
∂
P
z
M
+
w
N T
ρ
M
+
∂
z
where
M
M+
and
M
w
are the molecular weight of cations and water, respectively;
P
is the local osmotic fluid pressure,
P
f
, minus the local swelling pressure or stress,
σ
*, such that
P
=
P
f
-
σ
*,
