Biomedical Engineering Reference
In-Depth Information
11.5.2 Results and validation for the macroscale model
As proposed in [43] to perform numerical simulations it is convenient to rewrite
problem (11.4) in non dimensional form. First, the longest PLA chains that we con-
sider feature
x
N
=
600 monomers,the shortest
x
1
=
15 units and the entire spectrum
is subdivided into
N
=
40 classes. To this purpose, we select the polymer thickness
k
ρ
w
s
as the characteristic time,
D
w
=
L
=
10
μ
m as the characteristic length,
τ
=
cm
3
as
the density of pure water and non degraded PLA respectively. Then, without change
of notation, the non-dimensional form of Eq. (11.4) reads as follows,
⎧
⎨
10
−
7
cm
2
ρ
w
=
0
.
×
/
.
,
ρ
=
.
/
4
1
s
as reference diffusivity and finally
0
98
1
18
g
∂
t
ρ
w
=
Λ∇
·
D
∇ρ
w
−
N
i
=
1
i
−
1
i
ρ
w
ρ
i
,
K
∑
∂
t
ρ
i
=
Λ∇
D
i
∇ρ
i
−
(
N
j
i
j
ρ
j
,
i
−
1
)
x
1
ρ
w
ρ
i
+
2
x
1
ρ
w
∑
=
i
+
1
(11.5)
∂
z
ρ
w
|
z
=
0
=
0
, −D
w
∂
z
ρ
w
|
z
=
L
=
Γπ
w
(
ρ
w
|
z
=
L
−
A
)
⎩
∂
z
ρ
i
|
z
=
0
=
0
, −D
i
∂
z
ρ
i
|
z
=
L
=
Γπ
i
ρ
i
|
z
=
L
w
i
0
˜
ρ
w
(
z
,
0
)=
0
,
ρ
i
(
z
,
0
)=
ρ
where the diffusivity of water and polymer are determined by atomistic simulations,
as previously described, while the non-dimensional numbers
D
w
L
2
k
0
M
0
x
1
ρ
w
,
Γ
=
w
D
w
.
M
w
ρ
L
π
Λ
=
ρ
w
,
K
=
(11.6)
10
4
,where
k
10
−
2
day
−
1
is
taken from [43] and references therein, which confirms that PLA is a bulk erod-
ing polymer,
We estimate the Thiele number as
Λ
=
7
×
=
5
×
10
−
3
w
D
w
=
Γ
=
/
=
.
=
with the assumption
π
1,
K
0
0042 with
x
1
15,
M
0
mol
are the molecular weights of PLA and water respec-
tively. Finally, the saturation of water into dry polymer is estimated by [41] as
1
gwater
=
90
,
M
w
=
18
g
/
gPLA
,i.e.
A
0
5%.
For the estimation of the diffusivity of water and polymer with respect to the
degree of swelling
/
=
0
.
φ
w
and the average degree of polymerization
x
, we have applied
the static multiscale coupling strategy previously described. In particular, for water
we obtain,
2
w
D
w
(
φ
w
,
x
)=
−
3
.
524
+
92
.
974
φ
w
+
0
.
0137
x
−
0
.
115
φ
w
x
+
17
.
540
φ
0000213
x
2
2
w
x
φ
w
x
2
w
x
2
2
−
0
.
+
0
.
0916
φ
+
0
.
000183
−
0
.
000142
φ
.
For the diffusivity
D
i
we have considered a slightly different approach. Denoting
with
D
i
the data-set relative to Table 11.3 and with
D
i
the parametrized function to
be estimated, we have solved the following least square problem,
a
∗
=
D
i
−
arg mi
a
log
(
D
i
)
.