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

Himenz P.C. Polymer Chemistry. The Basic Concepts - Copyright, 1984. - 736 p.
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this shape are indeed observed. We shall see presently, however, that
this shape is also consistent with other mechanisms besides the one
considered until now.
In terms of spontaneous crystallization, the assumption that N
nuclei commence to grow simultaneously at t = 0 is the most unrealistic.
We can modify the model to allow for sporadic, spontaneous nucleation by
the following
The Kinetics of Crystallization: The Avrami Equation
223
argument. We draw a set of concentric rings in the plane of the disks
around point x as shown in Fig. 4.6b. If the radii are r and r + dr for
the rings, then the area enclosed between them is 2nr dr. We postulate
that spontaneous random nucleation occurs with a frequency N, having
units area-1 time-1. The rate of formation of nuclei within the ring is
therefore N27rr dr.
We continue to assume that the crystals so nucleated display a
constant rate of radial growth f. This means that it takes a crystal
originating in a ring of radius r around point x a time given by r/r to
cross x. The crystal labeled A in Fig. 4.6b has had just enough growth
time to reach x. On the other hand, a crystal nucleated in this ring
after t - r/r will not have had time to grow to x. The crystal labeled Â
in Fig. 4.6b is an example of the latter. It is only nucleation events
that occur up to t - r/r which have time to grow from the ring of radius
r and cross point x by their growth front. The increment in this number
of fronts for the ring of radial thickness dr is
dF = (N27rrdr)|t - *rj
(4.25)
The average number of fronts crossing point x at a time of
observation t is the sum of contributions from all rings which are within
reach of x in time t. The most distant ring included by this criterion is
a distance ft from x. The average number of fronts, therefore, is given
by integrating Eq. (4.25) for all rings between r = 0 and r = ft:
F = 2ttN Ä |t - v j dr
(4.26)
As far as this integration is concerned, f and t are constants, so Eq.
(4.26) is readily evaluated to give
F = - ;rNf2t3 (4.27)
3
As before, this quantity in relation to the degree of crystallinity is
given by Eq. (4.22), so equating the latter to Eq. (4.27) gives
1ï(^ò)=5^,Ý <4'28)
or
= 1 - exp (- J Nr2t3j
(4.29)
224
The Glassy and Crystalline States
Equations (4.24) and (4.29) are equivalent, except that the former
assumes instantaneous nucleation at N sites per unit area while the
latter assumes a nucleation rate of N per unit area per unit time. It is
the presence of this latter rate which requires the power of t to be
increased from 2 to 3 in this case.
In deriving these results we have focused attention on growth fronts
originating elsewhere and crossing point x. We would count the same
number if the growth originated at x and we evaluated the number of
nucleation sites swept over by the growing front. This change of
perspective is immediately applicable to a three-dimensional situation as
follows. Suppose we let N represent the number of sites per unit volume
(note that this is a different definition than given above) and assume
that a spherical growth front emanates from each. Then the average number
of fronts which cross nucleation sites in time t is
F = ~ ;r(rt)3N
(4.30)
by analogy with Eq. (4.18). Following the same argument as produced Eq.
(4.24), we obtain
â = 1 - exp (- | ÿã3Nt3J
(4.31)
for the three-dimensional case of simultaneous nucleation (concentration
N). Following the argument that produced (4.29), we obtain
â = 1 - exp ^ r3Nt4 j
(4.32)
for the three-dimensional case of sporadic nucleation (rate N).
Equations (4.24), (4.29), (4.31), and (4.32) all describe different
mechanisms for crystallization, yet all have basically the same form:
â = 1 - exp(-Ktm)
(4.33)
where Ê is a cluster-different in detail for each case-of numerical and
growth constants and m is the power dependence of time. Equation (4.33)
is known as the Avrami equation, after one of the researchers who has
made contributions in this area. Likewise, the exponent m in Eq. (4.33)
is called the Avrami exponent. The latter takes on the values 2, 3, or 4
for the four mechanisms considered above. To acquire some numerical
familiarity with the Avrami function, consider the following example.
The Kinetics of Crystallization: The Avrami Equation
225
Example 4.3
Three different crystallization systems show in values of 2, 3, and 4.
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