POLYMERS AS NATURAL COMPOSITES: STRUCTURE AND PROPERTIES - Fractal Mechanics of Polymers - page 335

h

= d - 1

, (15.36)

n

p

where ν p is correlation length index in percolation theory, d is dimension of
Euclidean space, in which a fractal is considered.
As it is known [4], the polymers nanocluster structure represents itself
the percolation system, for which p = φ cl , p c = 0.34 [35] and further it can be
written:

R

( )

n

cl

~ f - 0, 34

p

, (15.37)

cl

l

st

where R cl is the distance between nanoclusters, determined according to the
Eq. (4.63), l st is statistical segment length, ν p is correlation length index, ac-
cepted equal to 0.8 [77].
Since in the considered case the change E p at n cl variation is interesting
first of all, then the authors of Ref. [74] accepted cl = const = 2.5×10 27 m- 3 , l st
= const = 0.434 nm. The value E p calculation according to the Eqs. (15.35)
and (15.37) allows to determine this parameter according to the formula
[74]:

( ) (

d -1) n

E = 28, 9 f - 0, 34

p

, GPa. (15.38)

p cl

In Fig. 15.34 t he theoretical dependence (a solid line) of E p on nanoclu-
sters size (diameter) D cl , calculated according to the Eq. (15.38) is adduced.
As one can see, the strong growth E p at D cl decreasing is observed, which is
identical to the shown one i n Fig. 15.33 . The adduced in Fig. 15.34 experi-
mental data for REP, subjected to hydrostatic extrusion and subsequent an-
nealing, correspond well enough to calculation according to the Eq. (15.38).
The decrease D cl from 3.2 up to 0.7 nm results again to E p growth on order
of magnitude [74].
The similar effect can be obtained for linear amorphous polycarbonate
(PC) as well. Calculation according to the Eq. (15.38) shows, n cl reduction
from 16 (the experimental value n cl at T = 293K for PC [5]) up to 2 results
to E p growth from 1.5 up to 5.8 GPa and making of structureless ( n cl = 1) PC
will allow to obtain E p ≈ 9.2 GPa, that is, comparable with obtained one for
composites on the basis of PC.