Biomedical Engineering Reference
In-Depth Information
that combined with the previous expression gives,
w
D
w
.
D
i
π
π
i
=
The water saturation of the mixture is modelled with the partition factor
A
. The
non-degraded mixture is characterized by a partition factor 0
A
0
<
<
1 (that will be
later determined according to data taken by [18]). As erosion of the polymer at the
interface leads to a decrease in the total polymer density, denoted with ˜
ρ
=
∑
i
ρ
i
,we
propose the linear relationship
˜
ρ
|
z
=
L
ρ
w
A
0
A
=
1
−
(
1
−
)
such that
A
→
1when ˜
ρ
|
→
0, which at saturation equilibrium results in a mix-
z
=
L
ρ
w
→
ρ
w
and
ture characterized by
ρ
i
→
0 for all
i
=
1
,...,
N
, i.e. the network was
replaced by pure surrounding fluid.
We finally take into account hydrolysis. One common tool to describe the lo-
calization of the scission event along large chains is a scission probability density
function, which distinguishes the likelihood of scission among scission locations in
a chain of average degree of polymerization
x
i
, i.e. the relationships among
k
i
,
j
with
i
fixed and
j
1. Some frequently encountered scission probability den-
sity functions are: random, parabolic, and central scission (cf. [20, 21] for details).
Aliphatic polyesters degrade by passive hydrolysis and are usually characterized
by random scission events. Random scission is defined with a constent probability
density function along the length of the chain, i.e.
k
i
,
1
=
∈
1
,
2
,...,
i
−
k
i
,
2
=
···
=
k
i
,
i
−
1
, for all
i
N
. Considering
k
as the rate of hydrolysis of the particular type of poly-
meric bond, all
k
i
,
j
are given by
=
2
,...,
k
i
,
j
=
k x
1
,
for
i
=
2
,...,
N
and
j
=
1
,...,
i
−
1
.
Hydrolysis happens due to the presence of water, hence its rate depends on the par-
tial density
ρ
w
. Because water might not be homogeneously distributed over the net-
work, the rate of reaction depends implicitly on space and leads to inhomogeneous
degradation. Following similar studies (cf. [40]), random hydrolysis is a 1st order
reaction with water, i.e. the polymeric bond has reaction rate that follows a linear
relationship with partial density of water such that
k
k
=
k
(
ρ
w
)=
ρ
w
where
k
is a constent reaction rate. In such way, bilinear terms (each featuring
ρ
w
and one
ρ
i
) appear in the reaction terms of Eqs. (11.1) and (11.2). Autocatalysis,
i.e. when the rate of scission depends on the presence of residual monomer, is not
accounted with this constitutive specification.
