Biomedical Engineering Reference
In-Depth Information
6.4.2
N
UMERICAL
S
IMULATION
Having found an explicit equation for
v
(
r
,
t
), we can now carry out numerical sim-
ulations to compare the theoretical dynamic contraction of ionic polymeric gels in
an electric field with those of experiments. In order to compare the experimental
results and observations with the proposed dynamic model, a number of assumptions,
simplifications, and definitions is first made. Consider the ratio
W
(
t
)/
W
(0), where
W
(
t
) is the weight of the entire gel at time
t
, and
W
0
=
W
(0) is the weight of the gel
at time
t
= 0, just before the electrical activation. Thus,
t
r
()
−
()
=
0
()
∫
∫
WWt
0
2
πρ
v
r,t rdrdt
(6.84)
0
r
i
This can be simplified to
t
r
0
∫
∫
()
( )
()
1
Wt W
/
0 1
=−
W
−
2
πρ
v
r,t rdrdt
(6.85)
0
0
r
i
The initial weight of the gel is related to the initial degree of swelling
q
=
V
(0)/
V
p
,
where
V
(0) is the volume of the gel sample at
t
= 0 and
V
p
is the volume of the dry
polymer sample. Numerical simulations were carried out based on the assumptions
that the cross-section of the gel remains constant during contraction of the gel sample,
and that
ε
=
e
= 1.6
×
10
-19
C, T = 300 K
α
= 1,
D
= 80
µ
= 0.8
×
10
-3
Pa.s
ρ
= 1000 kg/m
3
b
= 2.55
10
-10
m
k
= 1.3807
×
×
10
-23
J/K
r
i
= 6.08
×
10
-10
m
r
0
=
r
i
q
(1/2)
q
= 25, 70, 100, 200, 256, 512, 750
The initial length and cross section of the sample are, respectively,
0
= 1 cm and
S
= 1 mm
2
. Electric field strength,
E
, = 5.8 V/mm.
The results of numerical simulation have been compared with the experimental
results of Gong and coworkers (1994a, 1994b) and Gong and Osada (1994) for
comparable cases and good agreement is observed, as depicted in figure 6.19.
Also, Asaka and Oguro (2000) present a model for the kinetics of bending of
IPMNC strips that closely resembles the preceding formulation. In brief, they con-
sider the water flux,
J
, of hydrated cations and cationic migration towards the cathode
side. Note that the imposition of an electric field causes the hydrated cations to
migrate by electrophoresis, electro-osmosis, and capacitive or ionic current. Thus,
a material flux can be defined such that
