Biomedical Engineering Reference
In-Depth Information
Table 11.4.
Diffusion coefficient (10
−
8
cm
2
/
s
) of PLA chains of length
x
i
into themselves
(columns) with respect to % of water content
φ
w
(rows)
φ
w
2 %
20 %
40 %
60 %
80 %
100 %
600
0.33
1.3
3.5
3.7
42.5
-
300
0.50
1.5
5.2
4.8
35.2
-
150
0.45
0.8
8.4
6.8
38.6
-
x
i
75
0.37
1.2
5.0
6.8
24.0
-
30
0.51
1.6
3.8
6.8
27.6
-
1
480
530
540
590
650
-
mer degradation, showing a behaviour similar to water transport. This consideration
allows us to assume that the diffusivity of polymer sub-fractions of length
x
i
into a
mixture of average degree of polymerization
x
x
i
may be very similar to the sub-
fraction of length
x
i
in itself. In practice, we conclude that
D
i
(
φ
w
,
=
x
i
)
D
i
(
φ
w
,
x
)
for
≤
≤
any
x
1
x
N
. This is equivalent to say that Table 11.4 represents the diffusivity
of polymer subfraction
x
i
in all possible water/polymer mixtures.
Finally, we must note that normal diffusion is not reached for the polymer diffu-
sivity, since the slope of log
x
presents values below 1 (with the exception of
PLA monomers, for which the normal diffusion is instead reached). In this regard it
is important to remind that normal diffusion behaviour holds only in the case that the
observation time (i.e. the simulation time) is large enough to allow the particles to
show uncorrelated motion. There are cases (i.e. simulation time) however, in which
the mean squared displacement is not linear in time, as in the case of Einstein's re-
lation, but displays a different power law, i.e.
MSD
(
MSD
(
t
))
t
n
. If the exponent
n
has a
value close to 1 it denotes the normal or Fickian diffusion, whereas it denotes linear
or ballistic motion if is close to 2. When 1
(
t
)
∼
<
n
<
2 the regime is called superdiffusive
motion whereas if
n
1 the regime is called subdiffusion or anomalous diffusion.
An example of this latter behaviour the model of the
ant in the labyrinth
, see [54],
where a particle (the ant) performs a random walk on a grid on which sites are ran-
domly blocked for diffusion (the labyrinth). In this case the motion of a particle in
a labyrinth is determined by the shape of the labyrinth itself and therefore it is not
a random walk. Similarly, for the diffusion of water into a polymer network, if the
simulation time is too short, the only motion captured is the very fast movement
of a given molecule in the void space between polymer chains, namely an individ-
ual hole. In this case the trajectories are affected by the hole dimension or more in
general by the microstructure of the material, contrarily to what the Fickian regime
would request. The usual effect of anomalous diffusion is to create a smaller slope
of the mean square displacement curve and the influence of this problem may ex-
tend up to several nanoseconds and increases with the size of the permeant molecule.
This is the case of the diffusive behaviour observed in the polymer fragments in our
systems.
<
