Chemistry Reference
In-Depth Information
M
( )
e =
∑
I
PP
ln
,
(4.46)
i
i
i
where
M
is the minimum number of
d-
dimensional cubes with side e, neces-
sary for all elements of structure coverage,
P
i
is the event probability that
structure point belongs to
i-
i-th element of coverage with volume e
d
.
In its turn, polymer structure entropy change D
S
, which is due to fluctua-
tion free volume
f
g
, can be determined according to the Eq. (4.21). Compari-
son of the Eqs. (4.21) and (4.46) shows that entropy change in the first from
them is due to
f
g
change probability and to an approximation of constant the
values
I
(e) and D
S
correspond to each other. Further polymer behavior at
deformation can be described by the following relationship [84]:
d
∑
d
i
D =-
Sc
l
=
-
1
,
(4.47)
I
j
1
where
c
is constant, l
I
is drawing ratio.
Hence, the comparison of the Eqs. (4.21), (4.46) and (4.47) shows that
polymer behavior at deformation is defined by change
f
g
exactly, if this pa-
rameter is considered as probabilistic measure. Let us remind, that
f
c
such
definition exists actually within the frameworks of lattice models, where this
parameter is connected with the ratio of free volume microvoids number
N
h
and lattice nodes number
N
(
N
h
/
N
) [49]. The similar definition is given and
for
P
i
in the Eq. (4.46) [86].
The value
d
I
can be determined according to the following equation [82]:
€
M
‚
∑
PP
()ln ()
e
e
„
…
i
i
„
…
d
=
lim
i
=
1
.
(4.48)
I
ln
e
„
…
e
→
0
„
…
†
‡
Since polymer structure is a physical fractal (multifractal), then for it
fractal behavior is observed only in a some finite range of scales (Fig. 1.2)
and the statistical segment length
l
st
is accepted as lower scale. Hence, as-
suming
P
i
(e) =
f
g
and e →
l
st
the Eq. (4.48) can be transformed into the fol-
lowing one [73]:
