Biomedical Engineering Reference
In-Depth Information
transport for simple geometries such as fi lms and cylinders. The concentration distance profi le of
concentrations (
C
) of undoped or doped PPy in an infi nite fi lm of thickness
fi
such that 0
<
<
x
l,
=
where
x
fi
is the surface of the fi lm that is exposed to solution, is given by
∞
C
−
C
0
π
∑
n
=
(
−
1)
n
+
1)
2
π
2
t
2
/4
fi
2
} cos
(2
n
1)π
x
4
________
C
1
______
2
n
__
_________
C
0
=
−
1
exp {
−
D
(2
n
+
1
2
fi
(13.2)
−
+
0
where
C
0
is the initial concentration,
C
1
the maximum concentration of counterions and water in
the fi lm, and
D
the effective diffusion coeffi cient for simultaneous transport of electronic charge,
counterions, and water.
29
In a cylinder of radius
a
such that 0
<
<
=
r
a
, where
r
a
is the cylinder
surface exposed to solution, the concentration profi le is given by the following:
∞
Dα
n
2
t
)
J
0
(
rα
n
)
C
−
C
0
a
∑
n
=
exp(
−
4
________
C
1
_________________
__
C
0
=
−
1
α
n
J
1
(
aα
n
)
(13.3)
−
1
where
J
0
and
J
1
are Bessel functions of orders zero and one, respectively, where
aα
n
are the roots
of the Bessel function of the fi rst kind of order zero.
The concentration profi les given in Equations 13.2 and 13.3 are plotted in Figure 13.26. The
current due to the concentration gradient can be determined by multiplying ∂
C
/∂
x
(
x
=
fi
) by
nFAD
,
where
A
is the effective area:
30
∞
Dt
)
−
exp
(
−
(
k
+
1)
2
fi
2
π
1/2
t
1/2
∑
k
nFAD
1/2
C
0
1)
k
[
exp
(
−
k
2
fi
2
)
]
__________
_________
___
=
−
i
i(l,t)
(
Dt
(13.4)
=
0
1.0
1.0
1.5
0.9
0.9
0.15
0.08
0.04
0.02
0.01
0.8
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.4
0.5
0.4
0.4
0.4
0.3
0.3
0.8
1.5
0.2
0.2
0.1
0
.08
0
.04
0.02
0.01
0.1
0.15
0.0
0.0
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
(B)
(A)
x/l
x/l
1.0
1.0
0.08
0.8
0.9
0.9
0.04
0.02
0.01
0.8
0.8
0.4
0.7
0.7
0.15
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.15
0.2
0.2
0.4
0.1
0.1
0
.
0
4
0
.
02
0.01
0.8
0.08
0.0
0.0
0
0.2
0.4
0.6
0.8
0
0.2
0.4
0.6
0.8
1
1
(C)
(D)
r/a
r/a
FIGURE 13.26
Concentration profi le versus distance at various times. The numbers on the curves are values of
Dt
/
fi
2
in (A) and (B) and
Dt
/
a
2
in (C) and (D). A fi lm of thickness
fi
is undergoing reduction with ions moving into
the fi lm from surface
x
=
fi
in (A), while (B) is the plot of the concentration profi le during oxidation (ion and water
movement out of the fi lm) in a fi lm. The same processes are plotted for a cylinder of radius
a
in (C) and (D).
