Biomedical Engineering Reference
In-Depth Information
with each scission reaction, i.e.
bond of scission
location
j
of chain
x
i
k
i
,
j
−→
x
1
·
+
H
2
O
scission event.
The rate
k
i
,
j
is the rate of reaction of scission location
j
composed of
x
1
bonds, hence
the rate of water consumption in the scission event is
k
i
,
j
/
x
1
. The rate of change of
the number of water molecules
n
w
=
n
w
(
x
,
t
)
in a representative volume d
V
is given
by
N
i
=
1
i
1
j
=
1
−
k
i
,
j
x
1
n
i
n
w
=
−
which, with the relationship between
n
w
and
d
V
(where
M
w
is the molecular mass of water), yields the water consumption due to the chemical
reaction in Eq.(6)
ρ
w
as d
V
→
0,
ρ
w
=
n
w
M
w
/
N
i
=
1
i
1
j
=
1
k
i
,
j
M
w
−
M
0
x
1
ρ
i
f
w
=
x
i
.
11.2.2 Initial and boundary value problem
Boundary and initial conditions depend on the application that one has in mind.
In our case, we aim to model the coating of a medicated stent, which is a thin
polymer layer covering the surface of a cardiovascular stent. For this reason, the
geometry of the polymer matrix can be thought of as a thin slab. Then, the gov-
erning equations can be reduced to one spatial dimension,
z
, the coordinate across
the thickness (
z
∈
[
0
,
L
]
where
L
is the coating thickness). The polymer network
starts out dry, i.e.
0 and the initial state of the polydisperse polymeric
system is homogeneous, i.e. independent of
z
, and is characterized by an initial de-
gree of polymerization distribution
w
0
ρ
w
(
z
,
0
)=
w
0
=
(
x
)
and an initial total (dry) polymer
0
density
ρ
. The initial weight fraction of chains of average degree of polymer-
ization
x
i
,
w
i
are obtained with Eq. (1) and the initial conditions of the polymeric
constituents densities are set as
w
i
ρ
0
ρ
i
(
z
,
0
)=
for
i
=
1
,...,
N
. In what follows,
indexes 0 and
∞
will refer to the state of the mixture at
t
→
0and
t
→
∞
, respec-
tively.
At
z
=
0 we apply impermeable boundary conditions
∂
z
ρ
w
|
z
=
0
=
0
,
∂
z
ρ
i
|
z
=
0
=
0
with the meaning that any constituent is not able to leave from the polymeric matrix
and penetrate the stainless steel bulk of the stent. At
z
=
L
, the polymer contacts sur-
rounding water or biological tissue. Water permeates through the interface according
to the following law,
−
D
w
∂
ρ
|
=
π
(
ρ
|
−
A
)
z
w
z
=
L
w
w
z
=
L
where
π
w
is the permeability of the interface to water molecules and A is a partition
