Biomedical Engineering Reference
In-Depth Information
Concerning the microscale model, we observe that, since the dimension of the
state space
ϒ
(
t
,
z
)=(
ρ
w
(
t
,
z
)
,
ρ
1
(
t
,
z
)
,...,
ρ
N
(
t
,
z
))
is considerably large, namely
N
1, it would be a challenging task to design an ensemble of atomistic simulations
sufficient to characterize constitutive laws of type
D
w
=
+
D
w
(
ρ
w
,
ρ
1
,...,
ρ
N
)
,
D
i
=
D
i
(
ρ
w
,
ρ
1
,...,
ρ
N
)
. By consequence, we first introduce a
lumped state space
, iden-
tified by a vector valued function
N
+
1
M
with
M
F
:
R
→
R
≤
N
+
1, such that it is
possible to initialize an atomistic PLA model on the basis of
solely. This as-
sumption will remarkably simplify the interacton between the micro and the macro
scale models. According to the results reported in the forthcoming sections, we select
a the follwing lumped state space with
M
F
(
ϒ
)
=
2:
1. the degree of swelling (also related to the extent of erosion),
ρ
(
t
,
z
)
w
F
1
(
ϒ
(
t
,
z
)) =
φ
w
(
ϒ
(
t
,
z
)) =
N
i
ρ
w
(
t
,
z
)+
∑
ρ
i
(
t
,
z
)
=
1
which quantifies the amount of water in the entire mixture;
2. the (weight) average degree of polymerization (the extent of degradation),
N
i
=
1
w
i
(
t
,
z
)
·
x
i
F
2
(
ϒ
(
t
,
z
)) =
x
(
ϒ
(
t
,
z
)) =
which quantifies the average size of polymeric chains in the mixture and always
satisfies
x
1
≤F
2
≤
x
N
.
We notice that this simplification involves some loss of information. On one hand,
according to degradation of polymeric chains, a polymer mixture is generally
poly-
disperse
, i.e. it is composed by a collection of several sub-fractions. On the other
hand, the selected lumped state space is not able to represent polydispersity and it
replaces a given distribution of sub-fractions with
monodisperse
mixture of equiv-
alent average degree of polymerization. Mathematically speaking, this corresponds
to say that function
1. In this framework, we sketch
below the input/output description of the microscale model:
F
is surjective, for any
M
<
N
+
(
φ
,
)
Input:
given
generate the corresponding atomistic model of PLA.
Compute:
select some tracers molecules (either water or polymer) to evaluate
molecular diffusion. Then, perform MD simulations to compute their trajectories
r
x
w
over a sufficiently large time span. Finally perform the mean square displace-
ment analysis combined with Einstein's relation in order to estimate the molecular
diffusivity
D
w
(
φ
w
,
(
t
)
x
)
,
D
i
(
φ
w
,
x
)
of the water or polymer tracers in a given PLA
mixture.
Output:
the model provides the molecular diffusivity of water and polymer
molecules, i.e. independent values
D
w
(
φ
w
,
x
)
,
D
i
(
φ
w
,
x
)
, into a given PLA/water
mixture.
A synthetic input/output relation for the microscale model reads as follows:
micro
=
⇒
F
(
ϒ
)
D
w
(
F
(
ϒ
))
,
D
i
(
F
(
ϒ
))
