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Polymer Chemistry. The Basic Concepts - Himenz P.C.

Himenz P.C. Polymer Chemistry. The Basic Concepts - Copyright, 1984. - 736 p.
Download (direct link): polymerchemistry1984.djvu
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Calculate the value required for in each of these systems so that all
will show = 0.5 after 103 sec. Use these m and values to compare the
development of crystallinity with time for these three systems.
Solution
Solve Eq. (4.33) for and evaluate at t = 103 sec for each of the m
values: = [-ln(l - 0)]/tm. For m = 2, = (In 0.5)/(103)2 = 6.93 X 10"7
sec"2; for m = 3, = 6.93 X "10 sec"3; for m = 4, = 6.93 X 10"13
sec"4. Note that the units of depend on the value of m. Solve Eq.
(4.33) for t and evaluate at different 0's for the m and values
involved:
m = 2
( = 6.93 X "7) (
0.1 3.89 X 102
0.2 5.67 X 102
0.3 7.18 X 102
0.4 8.59 X 102
0.5 1 X 103
0.6 1.15 X 103
0.7 1.32 X 103
0.8 1.52 X 103
0.9 1.82 X 103
t in seconds
m = 3 m = 4
= 6.93 X -10) (K = 6.93 X 10-13)
5.33 X 102 6.24 X 10:
6.85 X 102 7.53 X 10:
8.02 X 102 8.47 X 10:
9.03 X 102 9.27 X 10:
1 X 103 1 X 103
1.10X 103 1.07 X 10:
1.20 X 103 1.15 X 10:
1.32 X 103 1.23 X 10:
1.49 X 103 1.35 X 10:
These three systems describe a set of crystallization curves that cross
at = 0.5 and t = 103 sec. For the case where m = 2, the time span over
which the change occurs is widest (1430 sec from = 0.1-0.9) and the
maximum slope is flattest (7.8 X 10-4 sec-1 between = 0.4 and 0.6). For
m = 4, the range is narrowest (726 sec) and the maximum slope is steepest
(1.4 X 10-3 sec).

Although two of the mechanisms presented above yield the same power
dependence on t, it appears possible to eliminate certain mechanisms by
experimentally testing the development of with time. A strategy for
this is suggested by Eq. (4.28). Taking the logarithm of both sides of
that equation gives
In
In ' 1
1 -
= 3 In t + const. (4.34)
226
The Glassy and Crystalline States
or for the generalized Avrami equation
In
In
(rhr
= m In t + const. (4.35)
Thus the slope of a plot of ln[ln(l - 0)_1 ] versus In t will have a
slope equal to the Avrami exponent.
Before turning to an examination of this prediction, a few more
complications must be mentioned. Until now we have considered the case of
disk or spherical growth from simultaneous or sporadic nucleation. Are
these the only possibilities? As might be anticipated, the answer is no.
Other geometries for growth have been examined; we shall include only the
case of the cylinder, which is limited in radius but not in length (i.e.,
one-dimensional growth) to produce fibrillar structures. Another
modification which has been investigated concerns the rate-determining
step of the crystallization process. Equation (4.18) and those following
from it imply that a contact between the growing disk and the surrounding
melt for time t is sufficient for crystallization. Another possibility is
that allowance must be made for the diffusion of the molecules to (or
from) the growth site. A way of dealing with this assumes that amorphous
molecules must diffuse out of the crystal domain to allow space for the
crystallizing molecules. For a crystal of radius r, the time required for
molecules to diffuse out of this domain is, according to Eq. (2.63), r =
(2Dt//2. In Eq. (4.18) this radius is written r = rt. Thus, if the growth
rate is diffusion controlled, these two expressions for r can be equated
and solved for r:

(4.36)
If this result is substituted into the previous expressions containing r,
the effect is to replace r with (2D)1/2 and to multiply those t's which
accompany r by t~'/z.
This rather complex array of possibilities is summarized in Table
4.3. Table
4.3 lists the predicted values for the Avrami exponent for the following
cases:
1. Growth geometry: fibrillar rod, disk, and sphere.
2. Nucleation mode: simultaneous and sporadic.
3. Rate determination: contact and diffusion.
Those exponents which we have discussed explicitly are identified by
equation number in Table 4.3. Other tabulated results are readily
rationalized from these. For example, according to Eq. (4.24) for disk
(two-dimensional) growth on contact from simultaneous nucleations, the
Avrami exponent is 2. If the dimensionality of the growth is increased to
spherical (three dimensional), the exponent becomes 3. If, on top of
this, the mechanism is controlled by diffusion, the
The Kinetics of Crystallization: Experimental Aspects
227
Table 4.3 Summary of Exponents in the Avrami Equation for Different
Crystallization Mechanisms
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