Biomedical Engineering Reference
In-Depth Information
stituent yield the system of
N
reaction-diffusion equations
N
j
=
1
A
ij
ρ
j
,
for
i
=
1
,...,
N
∂
ρ
=
∇
·
(
D
i
∇ρ
)+
(11.1)
t
i
i
where
D
i
is the diffusivity of chains of average degree of polymerization
x
i
and co-
efficients
A
ij
are the reaction coefficients corresponding to chain cleavage due to
hydrolysis. More precisely, a chain of average degree of polymerization
x
i
can be
cleaved at
j
1 different scission locations (each composed of
x
1
individ-
ual polymeric bonds) to yield chains of smaller average degree of polymerization,
x
j
and
x
i
−
j
.All
i
=
1
,...,
i
−
−
1 possible outcomes of scission of a chain of degree of polymer-
ization
x
i
are
k
i
,
j
−→
x
i
x
j
+
x
i
−
j
,
for
i
=
1
,...,
N
,
j
=
1
,...,
i
−
1
.
The depolymerization kinetics of populations of individual molecules can be de-
scribed by means of a system of ordinary differential equations (cf. [33, 34]). Let
n
i
=
denote the number of chains of average degree of polymerization
x
i
exist-
ing at time
t
.Asd
V
n
i
(
t
)
→
0, the relationship between partial density
ρ
i
and number of
(
,
)=
(
,
)
/
molecules
n
i
is
d
V
,where
M
0
is the molecular mass of one
monomeric unit and
x
i
M
0
is the molecular weight of polymer subfractions of length
x
i
. In a closed system, the rate of change of
n
i
is given by,
ρ
x
t
n
i
x
t
x
i
M
0
i
i
1
j
=
1
k
i
,
j
n
i
+
−
N
∑
n
i
=
−
j
=
i
+
1
(
k
j
,
i
+
k
j
,
j
−
i
)
n
j
,
for
i
=
1
,...,
N
.
Then, Eq. (11.1) is complemented by the following expressions:
⎧
⎨
0
if
j
<
i
i
−
1
−
∑
m
=
1
k
i
,
m
if
j
=
i
A
ij
=
⎩
x
i
x
j
(
k
j
,
i
+
k
j
,
j
−
i
)
if
j
>
i
.
The combination of Eqs. (11.1) and (11.2) results in a multiscale description of poly-
mer degradation and erosion, as it combines a molecular description of chain scission
at the molecular level (second term on the right hand side of (11.1)) with macroscopic
Fick's law of diffusion (first term of the right hand side of (11.1)).
Water diffuses in the polymeric matrix and hydrolysis is accounted as a sink of
water, i.e.
∂
t
ρ
w
=
∇
·
(
D
w
∇ρ
w
)
−
f
w
(11.2)
where
ρ
w
=
ρ
w
(
x
,
t
)
and
D
w
are the partial density and the diffusivity of water. Reac-
tion term
f
w
>
0 accounts for water consumption: one water molecule is consumed
