Biomedical Engineering Reference
In-Depth Information
Curve fitting (5 V)
90
80
70
60
50
40
D = 17 gf/mm
30
20
10
0
0
0.5
1
1.5 2 2.5 3
Displacement of fiber (mm)
3.5
4
4.5
5
FIGURE 4.118
Force versus displacement, under 5 V.
ρ
*
is the charge density governed by the following Poisson's equation:
*
2
(4.19)
ρ
=−
D
*
∇
ψ
where
D
* is the dielectric constant of the liquid phase and
ψ
is governed by the
following Poisson-Boltzmann equation:
[
]
(4.20)
2
∇=
ψπε
(
nD
/
*) exp
−
εψ
/
T
where
n
is the number density of counter-ions
ε
is their average charge
k
is the Boltzmann constant
T
is the absolute temperature
The electrostatic potential in polyelectrolyte solutions for fully stretched mac-
romolecules is given by the following equation, which is an exact solution to the
Poisson-Boltzmann equation (4.20) in cylindrical coordinates:
{
}
(
)
−
2
2
2
2
−
1
ψ
(,)
rt
=
kT
ε
ln
r
/(
r
−
r
) sinh
β
ln
r r
/
tan
β
(4.21)
0
i
0
where
β
is related to
λ
= (
αε
2
/4
π
D
*
bkT
), where
α
is the degree of ionization (i.e.,
=
n
/
Z
, where
n
is the number of polyions and
Z
is the number of ionizable groups)
and
b
is the distance between polyions in the network. Furthermore,
n
=
[αε
α
2
/4
π
D
*
bkT
]
and
β
s are found from the following equation:
1
2
−
β
λ
=
(4.22)
(
)
1
+
β
coth
β
ln
r
i
0
