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
) .
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