Biomedical Engineering Reference
In-Depth Information
4.6.10
M
ATHEMATICAL
M
ODELING
OF
C
ONTRACTION
AND
E
LONGATION
OF
C-PAN F
IBERS
A possible explanation for the contraction and elongation is based upon the carbox-
ylic acid groups having the molecular geometry of activated PAN. At low pH
concentrations, all carboxylic acid groups on activated PAN are likely to be proto-
nated. This could potentially collapse the network by polymer-polymer affinity and
contract the polymer chain through neutral charge of the acid groups and hydrogen
bonding between neighboring carboxylic acid groups.
Based upon the Donnan theory of ionic equilibrium, the important forces arise
from (1) induced osmotic pressure of free ions between activated PAN fibers and
their environment; (2) ionic interaction of fixed ionic groups; and (3) the network
itself. Among these sources, the induced osmotic pressure of free ionic groups could
be the dominating force.
Electrical activation of PAN fibers is performed in an electrochemical cell such
as shown in figures 4.77 and 4.85. Note that, at the anode, oxygen evolves via 2H
2
O
⇒
H
2
+
2OH. Upon being hydrogenated in the vicinity of the PAN anode, the decreased pH
causes the PAN fibers to contract by the same effect as chemical activation. Also,
by reversing the polarity of DC, elongation of PAN fibers is simply obtained.
O
2
+ 4H
+
+ 4
e
+
and the counterreaction at the cathode is 2H
2
O + 2
e
-
⇒
4.6.10.1
Basic Modeling
Three important working forces that drive contraction or elongation of PAN muscles
were identified: rubber elasticity, proton pressure, and polymer-polymer affinity.
Rubber elasticity can be expressed by
Π
r
= -(
ρ
RTv
2
1/3
)/
M
c
(4.7)
where
Π
r
= contraction/elongation force by rubber elasticity
v
2
= volume fraction
T
= absolute temperature
ρ
= density of unswollen polymer
M
c
= molecular weight
R
= gas constant
The proton pressure is
∏=
e
(
ρ
RTf
/
M
)
v
(4.8)
c
2
where
f
is a number of dissociated hydrogen ions per chain.
The polymer-polymer affinity is
2
2
∏=
(
RT
/
V
)[
v
+
l (
1
−
v
)
+
xv
]
(4.9)
p
2
2
where
V
is the molar volume of the solvent and
x
is the Flory-Huggins parameter.
